#  Subgroups of a Subnormal Subgroup in Division Rings

Nguyen Anh Tu and Bui Xuan Hai111Corresponding author; e-mail address:
<EMAIL_ADDRESS>
Faculty of Mathematics and Computer Science, University of Science, Vietnam
National University - Ho Chi Minh City (VNU-HCMC), 227 Nguyen Van Cu Str.,
Dist. 5, HCM-City, Vietnam

Abstract

Let $D$ be a division ring with center $F$ and the multiplicative group
$D^{*}$. Our main purpose in this paper is to investigate the subgroup
structure of an arbitrary subnormal subgroup $G$ of $D^{*}$. In particular, we
prove that if $D$ is locally finite, then $G$ contains a noncyclic free
subgroup. The structure of maximal subgroups of $G$ is also investigated. The
new obtained results carry over some previous results about maximal subgroups
of $D^{*}$ to maximal subgroups of an arbitrary subnormal subgroup $G$ of
$D^{*}$.
Key words: division ring, maximal subgroup, subnormal subgroup, noncyclic free
subgroup, $FC$-element.
Mathematics Subject Classification 2010: 16K20, 16K40.

##  1 Introduction

Let $D$ be a division ring with center $F$ and $D^{*}$ be the multiplicative
group of $D$. The subgroup structure of $D^{*}$ is one of subjects which
attract the attention of many authors for last time. A particular interest is
that to study maximal subgroups of $D^{*}$, see for example, [1], [2], [4],
[8], [14], [21]. In this paper we replace $D^{*}$ by its arbitrary subnormal
subgroup $G$ and we study the subgroup structure of $G$. Recall that in [1],
[4], [21], Akbari et al. and Mahdavi-Hezavehi study maximal subgroups of
$D^{*}$ and many nice properties of such subgroups were obtained. In the
present paper, studying maximal subgroups of $G$, we get in many cases the
similar results for these subgroups as the results obtained in [1], [4], [21]
… for maximal subgroups of $D^{*}$. Other problem we study is the problem of
the existence of noncyclic free groups in a subnormal subgroups of a division
ring. This problem plays an important role for the understanding of the
structure of division rings. For more information we refer to the works
[5]-[7], [16]-[18], and [25]. Section 4 is devoted to this problem and the new
result we get in Theorem 4.4 gives the affirmative answer to Conjecture 2 from
[7] for the case of locally finite division rings.

In Section 2, we give some generalization of the well-known Wedderburn’s
Factorization theorem about polynomials over division rings. This
generalization and its corollaries will be used as important tools for our
study in the next sections of the paper. In the last part of this section, as
some illustration, we use them to generalize some previous results by Mahdavi-
Hezavehi et al. in [19].

Section 3 is devoted to the study of maximal subgroups of a subnormal subgroup
$G$ of $D^{*}$. Firstly, we consider the case when a maximal subgroup $M$ of
$G$ contains some non-central $FC$-element. The properties obtained in Theorem
3.5 are principal tools for our study in the next parts of the paper. Also, in
Theorem 3.7, we give a perfect description of the structure of $D$ when $M$ is
metabelian. In Section 5, we concentrate our study to the particular case when
$M$ is a maximal subgroup of $D^{*}$. The results we get are generalizations
of the result by Akbari et al. in [1, Theorem 6] and the result by B. X. Hai
and N. V. Thin in [8, Theorem 3.2] . More exactly, we replace the condition of
algebraicity of $M$ in [1, Theorem 6] and the condition of algebraicity of $D$
in [8, Theorem 3.2] by the weaker condition of algebraicity of derived
subgroup $M^{\prime}$ and the obtained results are the same. We refer to [2],
[10], and [14] for more information about the existence of maximal subgroups
in division rings.

Throughout this paper, for a ring $R$, the symbol $R^{*}$ denotes the group of
all units in $R$. For a non-empty subset $S$ of $D$, we denote by $F[S]$ and
$F(S)$ the subring and the division subring of $D$ respectively, generated by
the set $F\cup S$, where $F=Z(D)$ is the center of $D$. We say that a subgroup
$G$ of $D^{*}$ is absolutely irreducible if $F[G]=D$. Given a group $G$, a
subset $S$ and a subgroup $H$ of $G$, we denote by $Z(G)$, $G^{\prime}$,
$C_{G}(S)$ and $H_{G}=\bigcap_{x\in G}xHx^{-1}$ the center of $G$, the derived
group of $G$, the centralizer of $S$ in $G$ and the core of $H$ in $G$
respectively. If $x\in G$ and $H$ is a subgroup of $G$, then we denote by
$x^{H}$ the set of all elements $x^{h}=hxh^{-1},h\in H$. If $x^{G}$ is finite,
then we say that $x$ is an $FC$-element of $G$. The set of all $FC$-elements
of $G$ is called the $FC$-center of $G$. If $x,y\in G$ then
$[x,y]=xyx^{-1}y^{-1}$ and $[H,K]$ is the subgroup of $G$ generated by all
elements $[h,k],h\in H,\,k\in K$. We say that an element $x\in D$ is radical
over $F$ if there exists a positive integer $n(x)$ depending on $x$ such that
$x^{n(x)}\in F$. A non-empty subset $S$ of $D$ is radical over $F$ if every
element of $S$ is radical over $F$. We write $H\leq G$ and $H<G$ if $H$ is a
subgroup and proper subgroup of $G$ respectively. All other notation and
symbols in this paper are standard and one can find, for example, in [15],
[22], [23], [26], [27].

##  2 Algebraicity over a division subring and the generalization of
Wedderburn’s Factorization theorem

Let $D$ be a division ring with center $F$, and let $A$ be a conjugacy class
of $D$ which is algebraic over $F$ with minimal polynomial $f(t)\in F[t]$ of
degree $n$. Then, there exist $a_{1},\ldots,a_{n}\in A$ such that

$f(t)=(t-a_{1})\ldots(t-a_{n})\in D[t].$

This factorization theorem due to Wedderburn (see [15, (16.9), p. 265]) plays
an important role in the theory of polynomials over a division ring and its
applications in different problems are well-known. In this section, firstly we
consider the similar question in more general circumstance in order to get the
analogous theorem which could be used to generalize some previous results of
other authors about subgroups in division rings.

Let $K\subseteq D$ be division rings and $\alpha\in D$. We say that $\alpha$
is (right) algebraic over $K$ if there exists some nonzero polynomial $f(t)\in
K[t]$ having $\alpha$ as a right root. A monic polynomial from $K[t]$ with
smallest degree having $\alpha$ as a right root is called a minimal polynomial
of $\alpha$ over $K$. Throughout this paper we consider only right roots and
right algebraicity, so we shall always omit the prefix “right”. The minimal
polynomial of $\alpha$ over $K$ is unique, but it may not be irreducible as
the following example shows:

Let $\mathbb{H}$ be the division ring of real quaternions. Then,
$f(t)=t^{2}+1\in\mathbb{C}[t]$ is the minimal polynomial of $j$ and $k$ over
$\mathbb{C}$. Here, $\\{1,i,j,k\\}$ is standard basis of $\mathbb{H}$ over
$\mathbb{R}$.

The proof of the following lemma is a simple modification of the proof of
Lemma (16.5) in [15], so we now omit it.

######  Lemma 2.1.

Let $R$ be a ring, $D$ a division subring of $R$ and suppose $M$ is a subgroup
of $R^{*}$ normalizing $D^{*}$. If $K=C_{D}(M)$ and $x\in D^{*}$ is algebraic
over $K$ with the minimal polynomial $f(t)\in K[t]$, then a polynomial
$h(t)\in D[t]$ vanishes on $x^{M}$ if and only if $h(t)\in D[t]f(t)$.

Using Lemma 2.1, we get the following theorem which can be considered as some
generalization of Wedderburn’s Factorization theorem.

######  Theorem 2.2.

Let $R$ be a ring, $D$ a division subring of $R$ and suppose $M$ is a subgroup
of $R^{*}$ normalizing $D^{*}$. If $K=C_{D}(M)$ and $x\in D^{*}$ is algebraic
over $K$ with the minimal polynomial $f(t)$ of degree n, then there exist
$x_{1},\ldots,x_{n-1}\in x^{MD^{*}}$ such that

$f(t)=(t-x_{n-1})\cdots(t-x_{1})(t-x)\in D[t].$

###### Proof.

Take a factorization

$f(t)=g(t)(t-x_{r})\cdots(t-x_{1})(t-x)$

with $g(t)\in D[t]$, $x_{1},\ldots,x_{r}\in x^{MD^{*}}$, where $r$ is chosen
as large as possible. We claim that $h(t):=(t-x_{r})\cdots(t-x_{1})(t-x)$
vanishes on $x^{M}$. In fact, consider an arbitrary element $y\in x^{M}$. If
$h(y)\neq 0$, then, by [15, (16.3), p. 263], $g(x_{r+1})=0$, where
$x_{r+1}=aya^{-1}\in x^{MD^{*}},a=h(y)$. It follows that
$g(t)=g_{1}(t)(t-x_{r+1})$ for some $g_{1}(t)\in D[t]$, and so

$f(t)=g_{1}(t)(t-x_{r+1})(t-x_{r})\cdots(t-x_{1})(t-x).$

Since this contradicts to the choice of $r$, we have $h(x^{M})=0$; so, in view
of Lemma 2.1, $r=n-1$. Hence, $f(t)=(t-x_{n-1})\cdots(t-x_{1})(t-x)$, as it
was required to prove. ∎

We note that, the Wedderburn’s Factorization theorem is the special case of
Theorem 2.2, when we take $R=D$ and $M=D^{*}$.

From Theorem 2.2 and [15, (16.3), p. 263], we get the following corollaries.

######  Corollary 2.3.

Let $R$ be a ring, $D$ a division subring of $R$ and suppose $M$ is a subgroup
of $R^{*}$ normalizing $D^{*}$. Assume that $K=C_{D}(M)$ and $x\in D^{*}$ is
algebraic over $K$ with the minimal polynomial $f(t)$. If $y$ is a root of
$f(t)$ in $D$, then $y\in x^{MD^{*}}$.

######  Corollary 2.4.

Let $R$ be a ring, $D$ a division subring of $R$ and suppose $M$ is a subgroup
of $R^{*}$ such that $D^{*}\trianglelefteq M$. Assume that $K=C_{D}(M)$ and
$x\in D^{*}$ is algebraic over $K$ with the minimal polynomial $f(t)$ of
degree n. Then, $K$ is contained in the center of $D$ and there exists an
element $c_{x}\in[M,x]\cap K(x)$ such that $x^{n}=N_{K(x)/K}(x)c_{x}$ with
$N_{K(x)/K}(c_{x})=1$, where $N_{K(x)/K}$ is the norm of $K(x)$ to $K$.

###### Proof.

Since $D^{*}\leq M$, $K$ is contained in the center of $D$ and $K(x)$ is a
field. Taking $b=N_{K(x)/K}(x)$, by Theorem 2.2 we have $b=x^{r_{1}}\cdots
x^{r_{n}}$, with $r_{1},\ldots,r_{n}\in M$. We can write $b$ in the following
form:

$b=[r_{1},x][r_{2},x]^{x}[r_{3},x]^{x^{2}}\cdots[r_{n},x]^{x^{n-1}}x^{n}.$

Putting

$c^{-1}_{x}=[r_{1},x][r_{2}^{x},x][r_{3}^{x^{2}},x]\cdots[r_{n}^{x^{n-1}},x],$

we have $c_{x}=b^{-1}x^{n}\in[M,x]\cap K(x)$. So,

$N_{K(x)/K}(c_{x})=N_{K(x)/K}(b^{-1})N_{K(x)/K}(x)^{n}=b^{-n}b^{n}=1,$

as it was required to prove. ∎

This corollary can be reformulated in the following form which should be
convenient in some cases of its application in the next sections.

######  Corollary 2.5.

Let $R$ be a ring, $D$ a division subring of $R$ and suppose $M$ is a subgroup
of $R^{*}$ normalizing $D^{*}$. Assume that $K=C_{D}(M)$ is a field and $x\in
Z(D)^{*}\cap M$ is algebraic over $K$ with the minimal polynomial $f(t)$ of
degree n. Then, there exists an element $c_{x}\in[M,x]\cap K(x)$ such that
$x^{n}=N_{K(x)/K}(x)c_{x}$ with $N_{K(x)/K}(c_{x})=1$.

###### Proof.

Let $b=N_{K(x)/K}(x)\in K$, by Theorem 2.2, we have $b=x^{r_{1}d_{1}}\cdots
x^{r_{n}d_{n}}$, with $r_{1},\ldots,r_{n}\in M$ and $d_{1},\ldots,d_{n}\in
D^{*}$. We can write $b$ in the following form:

$b=[r_{1}d_{1},x][r_{2}d_{2},x]^{x}[r_{3}d_{3},x]^{x^{2}}\cdots[r_{n}d_{n},x]^{x^{n-1}}x^{n}.$

Since $x\in Z(D)^{*}$, we get

$b=[r_{1},x][r_{2},x]^{x}[r_{3},x]^{x^{2}}\cdots[r_{n},x]^{x^{n-1}}x^{n}.$

Putting

$c^{-1}_{x}=[r_{1},x][r_{2}^{x},x][r_{3}^{x^{2}},x]\cdots[r_{n}^{x^{n-1}},x],$

we have $c_{x}=b^{-1}x^{n}\in[M,x]\cap K(x)$. So,

$N_{K(x)/K}(c_{x})=N_{K(x)/K}(b^{-1})N_{K(x)/K}(x)^{n}=b^{-n}b^{n}=1,$

as it was required to prove. ∎

In view of Corollary 2.4, the following fact is evident.

######  Corollary 2.6.

Let $R$ be a ring, $D$ a division subring of $R$ and suppose $M$ is a subgroup
of $R^{*}$ such that $D^{*}\trianglelefteq M$. If $K=C_{D}(M)$, then
$K\subseteq Z(D)$ and the following assertions hold:
(i) If D is algebraic over K, then $D^{*}/K^{*}[M,D^{*}]$ is a torsion group.
(ii) If $[D:K]<\infty$, then $D^{*}/K^{*}[M,D^{*}]$ is a torsion group of a
bounded exponent dividing $[D:K]$.

For the illustration of the use of the generalization of Wedderburn’s
Factorization theorem, we are now ready to get the following result, which can
be considered as a generalization of Theorem 1 in [19].

######  Theorem 2.7.

Let $R$ be a ring, $D$ a division subring of $R$ and suppose $M$ is a subgroup
of $R^{*}$ such that $D^{*}\trianglelefteq M$. If $K=C_{D}(M)$ and $H/K$ is a
finite separable field extension in $D$, then the following statements hold:
(i) There exists an element $c\in[M,H^{*}]$ such that $N_{H/K}(c)=1$ and
$H=K(c)$. (ii) Either $H$ is algebraic over the prime subfield
$\mathbb{Z}_{p}$ or there exists an element $c\in[M,H^{*}]$ such that
$N_{H/K}(c)=1$ and $H=K(c)=K(c^{2})=K(c^{3})=\cdots$. (iii) If $H$ has a
nontrivial $K$-automorphism, then there exists an element $z=[m,h]$ with $m\in
M,\ h\in H^{*}$ such that $N_{H/K}(z)=1$ and $H=K(z)$.

###### Proof.

The proof of this theorem is similar to the proof of Theorem 1 in [19]. To get
the proof of our theorem, it suffices to replace the use of Skolem-Noether
theorem and Lemma A in the proof of Theorem 1 in [19] by the use of Corollary
2.3 and Corollary 2.4 respectively. ∎

##  3 Maximal subgroups of subnormal subgroups

The main purpose in this section is to describe the structure of maximal
subgroups in an arbitrary subnormal subgroup of $D^{*}$ and show their
influence to the whole structure of $D$. In the first, we prove the following
useful lemmas we need for further study.

######  Lemma 3.1.

If $D$ is a division ring with center $F$ and $G$ is a solvable-by-locally
finite subnormal subgroup of $D^{*}$, then $G\subseteq F$.

###### Proof.

By assumption there is a solvable normal subgroup $S$ of $G$ such that $G/S$
is locally finite. So, $S$ is a solvable subnormal subgroup of $D^{*}$ and in
view of [23, 14.4.4, p. 440], we have $S\subseteq F$. Hence, $G/Z(G)$ is
locally finite, so by [1, Lemma 3], $G^{\prime}$ is locally finite. Thus,
$G^{\prime}$ is a torsion subnormal subgroup of $D^{*}$ and by [12, Theorem
8], $G^{\prime}\subseteq F$. Hence, $G$ is solvable and by [23, 14.4.4, p.
440], $G\subseteq F$. ∎

######  Lemma 3.2.

Let $D$ be a division ring with center $F$ and $G$ a subnormal subgroup of
$D^{*}$. If $G$ is locally polycyclic-by-finite (e.g. if $G$ is locally
nilpotent) then $G\subseteq F$.

###### Proof.

If $G\not\subseteq F$, then by Stuth’s theorem (see, for example, [23, 14.3.8,
p. 439]) we have $D=F(G)$. By a result of Lichtman [24, 4.5.2, p. 155]
together with an exercise in [24, p. 162] or with the fact that $G$ is
contained in a unique maximal locally polycyclic-by-finite normal subgroup of
$D^{*}$, it follows that $G$ contains a noncyclic free subgroup, but this
contradicts to the fact that $G$ is locally polycyclic-by-finite. ∎

######  Lemma 3.3.

Let $D$ be a division ring with center $F$, $G$ a subnormal subgroup of
$D^{*}$ and suppose that $M$ is a maximal subgroup of $G$. If
$[M:Z(M)]<\infty$, then $M$ is abelian.

###### Proof.

Suppose that $M$ is non-abelian. Let $\\{x_{1},\ldots,x_{t}\\}$ be a
transversal of $Z(M)$ in $M$, $E$ a subfield of $D$ generated by $Z(M)$ and
$K=E[M]=\\{\sum_{i=1}^{t}a_{i}x_{i}|a_{i}\in E\\}$. Then, $K$ is a finite
dimensional vector space over $E$ and this implies that $K$ is a division
ring. Clearly, $E\subseteq Z(K)$, so we have $[K:Z(K)]<\infty$ and
$C_{K}(M)=Z(K)$. Since $M$ is maximal in $G$ and $M$ normalizes $K^{*}$, it
follows that either $G$ normalizes $K^{*}$ or $K^{*}\cap G\subseteq M$.
Case 1: $G$ normalizes $K^{*}$.
By Stuth’s theorem, either $K\subseteq F$ or $K=D$. If $K\subseteq F$ then
$M\subseteq F$, a contradiction. Thus, $D=K$; hence $Z(M)\subseteq
C_{D}(M)=Z(D)=F$ and, consequently, $Z(M)=F^{*}\cap M$. Take an $x\in
G\setminus M$ and put $H=\langle x,x_{1},\ldots,x_{t}\rangle$. By maximality
of $M$ in $G$, we get $G=HZ(M)=H(F^{*}\cap M)$. It follows that
$G^{\prime}\leq H$, so $H$ is normal in $G$. By [20, Theorem 1], we have
$H\subseteq F$, hence $G\subseteq F$. In particular, $M$ is abelian, a
contradiction.
Case 2: $K^{*}\cap G\subseteq M$.
In this case $K^{*}\cap G=M$, so $M$ is a subnormal subgroup of $K^{*}$. For
$N=\langle x_{1},\ldots,x_{t}\rangle$, we have $M=NZ(M)$, hence
$M^{\prime}\leq N$. Thus, $N$ is a finitely generated subnormal subgroup of
$K^{*}$. Recall that $[K:Z(K)]<\infty$, so in view of [20, Theorem 1], $N$ is
abelian and, consequently $M$ is abelian, a contradiction. This completes our
proof. ∎

It is well-known that the conjugacy class of a non-central element in $D$ has
the same cardinality with $D$ (see, for example, [23, 14.2.2, p. 429]). This
fact can be generalized by the following theorem which should be useful for
the next study.

######  Theorem 3.4.

If $D$ is a division ring with center $F$ and $G$ is a non-central subnormal
subgroup of $D^{*}$, then $|x^{G}|=|D|$ for any $x\in D\setminus F$.

###### Proof.

We claim that $|x^{G}|$ is infinite for any $x\in D\setminus F$. Thus, assume
that $|x^{G}|<\infty$, or, equivalently that $[G:C_{G}(x)]<\infty$. Since $G$
normalizes $F(x^{G})$, by Stuth’s theorem it follows that $F(x^{G})=D$.
Putting $H=(C_{G}(x))_{G}$, we have $H\subseteq C_{D}(x^{G})=C_{D}(D)=F$. On
the other hand, by $[G:C_{G}(x)]<\infty$, $[G:H]<\infty$, so $G\subseteq F$ by
Lemma 3.1, a contradiction. Therefore $|x^{G}|$ is infinite. Let $R$ be a
division ring generated by $x^{G}$, by Stuth’s theorem, $R=D$, so
$|x^{G}|=|D|$. ∎

Using Theorem 3.4, in the following we show the role of an $FC$-element in
maximal subgroups of a subnormal subgroup of $D^{*}$. The results obtained in
this theorem will be used as principal tools in the proofs of next theorems.

######  Theorem 3.5.

Let $D$ be a division ring with center $F$, $G$ a subnormal subgroup of
$D^{*}$ and suppose that $M$ is a maximal subgroup of $G$, containing a non-
central $FC$-element $\alpha$. By setting $K=F((\alpha)^{M})$ and
$H=C_{D}(K)$, the following conditions hold:
(i) $K$ is a field, $[K:F]=[D:H]<\infty$ and $F(M)=D$. (ii) $K^{*}\cap G$ is
the $FC$-center and also the Fitting subgroup of $M$. (iii) $K/F$ is a Galois
extension, $M/H^{*}\cap G\cong\mathrm{Gal}(K/F)$ is a finite simple group.
(iv) If $H^{*}\cap G\subseteq K$, then $H=K$ and $[D:F]<\infty$.

###### Proof.

First, we claim that $F(M)=D$. In fact, if $F(M)^{*}\cap G=M$, then $M$ is a
subnormal subgroup of $F(M)^{*}$ containing the non-central $FC$-element
$\alpha$, so we have a contradiction in view of Theorem 3.4. Thus,
$F(M)^{*}\cap G=G$ by maximality of $M$ in $G$; hence, by Stuth’s theorem, we
have $F(M)=D$.

Since $\alpha$ is an $FC$-element of $M$, $[M:C_{M}(\alpha)]<\infty$. Taking
$N=(C_{M}(\alpha))_{M}$, $K=F(\alpha^{M})$ and $H=C_{D}(K)$, we have
$N\vartriangleleft M$, $N\leq H^{*}$ and $[M:N]$ is finite. Since $M$
normalizes $K^{*}$, by maximality of $M$ in $G$, it follows that, either $G$
normalizes $K^{*}$ or $K^{*}\cap G\leq M$. If $G$ normalizes $K^{*}$, then by
Stuth’s theorem we have $K=D$, so $H=F$. Therefore, $N\leq F^{*}\cap M\leq
Z(M)$. The inequality $[M:C_{M}(\alpha)]<\infty$ implies $[M:N]<\infty$ and
consequently, $[M:Z(M)]<\infty$. Now, in view of Lemma 3.3, $M$ is abelian, a
contradiction. Thus $K^{*}\cap G=K^{*}\cap M\trianglelefteq M$, hence
$K^{*}\cap G$ is a subnormal subgroup of $K^{*}$ containing the set
$\alpha^{M}$ of $FC$-elements in $M$. By Theorem 3.4, $\alpha^{M}\subseteq
Z(K)$, hence $K$ is a field. Since $H=C_{D}(K)$ and $M$ normalizes $K^{*}$,
$M$ also normalizes $H^{*}$. By maximality of $M$ in $G$, either $G$
normalizes $H^{*}$ or $H^{*}\cap G\leq M$. If $G$ normalizes $H^{*}$, then by
Stuth’s theorem, either $H=D$ or $H\subseteq F$. However, both cases are
impossible since $K\subseteq C_{D}(K)=H$ and $K$ contains an element
$\alpha\not\in F=Z(D)$. Therefore, $H^{*}\cap G=H^{*}\cap M\trianglelefteq M$.
Since $N\leq H^{*}\cap G$ and $[M:N]<\infty$, we have $[M:H^{*}\cap G]<\infty$
and $M=\bigcup_{i=1}^{t}x_{i}(H^{*}\cap G)$ for some transversal
$\\{x_{1},\ldots,x_{t}\\}$ of $H^{*}\cap G$ in $M$. Put
$R=\sum_{i=1}^{t}x_{i}H$. Since $M$ normalizes $H^{*}$, it is easy to see that
$R$ is a ring and also, it is a finite-dimensional right vector space over its
division subring $H$. This fact implies that $R$ is a division subring of $D$
containing $F(M)$. Therefore $R=D$ and $[D:H]<\infty$. By using Double
Centralizer theorem [15, 15.4, p. 253] we get $[K:F]=[D:H]<\infty$ and
$Z(H)=C_{D}(H)=K$. Thus $(i)$ is established.
Since $M$ normalizes $K^{*}$, for all $a\in M$ the mapping $\theta_{a}:K\to K$
given by $\theta_{a}(x)=axa^{-1}$ is an automorphism of $K/F$. Now, consider
the mapping $\psi:M\to\mathrm{Gal}(K/F)$ given by $\psi(a)=\theta_{a}$.
Clearly, $\psi$ is a group homomorphism with
$\mathrm{ker}\psi=C_{M}(K^{*})=C_{D}(K)^{*}\cap M=H^{*}\cap M=H^{*}\cap G$.
Since $F(M)=D$, we obtain $C_{D}(M)=F$ and it follows that the fixed field of
$\psi(M)$ is $F$. By Galois correspondence, we conclude that $\psi$ is a
surjective homomorphism and $K/F$ is a Galois extension. Hence, $M/H^{*}\cap
G\cong\mathrm{Gal}(K/F)$ is a finite group. Assume that $\mathrm{Gal}(K/F)$ is
not simple. Then, there exists some intermediate subfield $L,F\subset L\subset
K$, such that $\theta(L)=L$ for all $\theta\in\mathrm{Gal}(K/F)$. Thus, $L$
and $E=C_{D}(L)$ are normalized by $M$ and clearly, $E\neq D$ and
$E\not\subseteq F$. Therefore, by Stuth’s theorem, $E^{*}$ is not normalized
by $G$. If $E^{*}\cap G\not\subseteq M$, then $G=M(E^{*}\cap G)$ normalizes
$E^{*}$, a contradiction. Hence, $E^{*}\cap G\leq M$. Since $K^{*}\cap
G\trianglelefteq M$ and $K^{*}\cap G\leq E^{*}\cap G$, we have $K^{*}\cap
G\trianglelefteq E^{*}\cap G$ and so $K^{*}\cap G$ is an abelian subnormal
subgroup of $E^{*}$. By Lemma 3.1, we have $K^{*}\cap G\subseteq Z(E)=L$.
Hence, $K=F(\alpha^{M})\subseteq F(K^{*}\cap G)\subseteq L$, a contradiction.
Thus $\mathrm{Gal}(K/F)$ is simple, and the proof of $(iii)$ is complete.
If $H^{*}\cap G\subseteq K$, then $H^{*}\cap G=K^{*}\cap G$. Recall that
$[M:H^{*}\cap G]<\infty$, hence $[M:K^{*}\cap G]<\infty$. Suppose that
$\\{y_{1},\ldots,y_{l}\\}$ is a transversal of $K^{*}\cap G$ in $M$. Since $M$
normalizes $K^{*}$, $Q=\sum_{i=1}^{l}y_{i}K$ is a division ring. Clearly, $Q$
contains both $F$ and $M$, so in view of $(i)$, we have $Q=D$. It is easy to
see that $[D:K]=[Q:K]\leq|M/H^{*}\cap G|=|\mathrm{Gal}(K/F)|=[K:F]$. Hence, by
Double Centralizer theorem, $K$ is a maximal subfield of $D$, $K=H$ and
$[D:F]<\infty$. Thus $(iv)$ is established.
To prove $(ii)$, first we claim that $K^{*}\cap G$ is a maximal abelian normal
subgroup of $M$. Thus, suppose $C$ is a maximal abelian normal subgroup of $M$
containing $K^{*}\cap G$. Then, $\alpha^{M}\subseteq C$ and it follows $C\leq
H^{*}$. Recall that $H^{*}\cap G\leq M$, so we have $C\trianglelefteq
H^{*}\cap G$. Hence, $C$ is an abelian subnormal subgroup of $H^{*}$, and by
Lemma 3.1, $C\subseteq Z(H)=K$. So $K^{*}\cap G=C$ is a maximal abelian normal
subgroup of $M$.
To prove $K^{*}\cap G$ is the Fitting subgroup of $M$, it suffices to show
that $K^{*}\cap G$ is a maximal nilpotent normal subgroup of $M$. Now, assume
that $A$ is a nilpotent normal subgroup of $M$ which strictly contains
$K^{*}\cap G$. Then, $B=H^{*}\cap G\cap A$ is a nilpotent subnormal subgroup
of $H^{*}$. Hence, by Lemma 3.1, we conclude that $B\subseteq Z(H)=K$ and
consequently, $B=K^{*}\cap G\cap A$. If $A\subseteq H^{*}\cap G$, then
$A=B\subseteq K^{*}\cap G$, a contradiction. Therefore, $A\not\subseteq
H^{*}\cap G$. Thus, $A(H^{*}\cap G)/H^{*}\cap G$ is a nontrivial normal
subgroup of $M/H^{*}\cap G$. Since $M/H^{*}\cap G$ is simple, $M/H^{*}\cap
G=A(H^{*}\cap G)/H^{*}\cap G\cong A/B$. Suppose $S=\sum_{i=1}^{m}z_{i}K$,
where $\\{z_{1},\ldots,z_{m}\\}$ is a transversal of $B$ in $A$. Since $A$
normalizes $K^{*}$ and $B\subseteq K$, $S$ is a division ring and
$[S:K]_{r}\leq m$. Recall that $M$ normalizes $A$ and $K$; so, $M$ also
normalizes $S$. By maximality of $M$ in $G$, it follows either $G$ normalizes
$S$ or $S^{*}\cap G\leq M$. If the second case occurs, then $A$ is a nilpotent
subnormal subgroup of $S^{*}$. By Lemma 3.1, $A$ is abelian and this
contradicts to the fact that $K^{*}\cap G$ is a maximal abelian normal
subgroup of $M$. Thus, $G$ normalizes $S$, and by Stuth’s theorem we have
$S=D$. Therefore, $[D:K]_{r}\leq m=|M/H^{*}\cap G|=|\mathrm{Gal}(K/F)|=[K:F]$.
This implies that $[D:F]=m^{2}$ and $K=H$ is a maximal subfield of $D$. From
$M/H^{*}\cap G=A(H^{*}\cap G)/H^{*}\cap G$ and $K^{*}\cap G<A$, it follows
$M=A$. Since $[D:F]<\infty$, $M$ can be considered as a nilpotent linear group
no containing unipotent elements ($\neq 1$). By a result in [3, p. 114],
$[M:Z(M)]$ is finite, which contradicts to Lemma 3.3. Thus, $K^{*}\cap G$ is
the Fitting subgroup of $M$.
For any $x\in K^{*}\cap G$, the elements of $x^{M}\subseteq K$ have the same
minimal polynomial over $F$, so $|x^{M}|<\infty$. For any $x\in M\setminus
K^{*}\cap G$, if $|x^{M}|<\infty$ then, by what we proved, $F(x^{M})\cap G$ is
the Fitting subgroup of $M$ which is different from $K^{*}\cap G$, a
contradiction. Hence, $|x^{M}|=\infty$ and $(ii)$ follows. Thus, the proof of
the theorem is now complete. ∎

From Theorem 3.5 we get the following corollary which is more convenient for
further applications.

######  Corollary 3.6.

Let $D$ be a division ring with center $F$, $G$ a subnormal subgroup of
$D^{*}$ and suppose $M$ is a maximal subgroup of G. If $M$ contains an abelian
normal subgroup $A$ and an element $\alpha\in A\setminus Z(M)$ algebraic over
$F(Z(M))$, then $K=F(A)$ and $H=C_{D}(K)$ satisfy (i) - (iv) in Theorem 3.5.
Moreover, $F(A)=F[A]$.

###### Proof.

Since $A\trianglelefteq M$, the elements of $\alpha^{M}$ in the field
$F(Z(M)A)$ have the same minimal polynomial over $F(Z(M))$. Hence,
$|\alpha^{M}|<\infty$. Now, by using Theorem 3.5, $K=F(\alpha^{M})$ and
$H=C_{D}(K)$ satisfy (i) - (iv) in Theorem 3.5. By (ii), $A\subseteq K$ and
$K=F(A)=F[A]$ since $[K:F]<\infty$. ∎

######  Theorem 3.7.

Let $D$ be a division ring with center $F,G$ a subnormal subgroup of $D^{*}$
and suppose $M$ is a non-abelian metabelian maximal subgroup of $G$. Then
$[D:F]=p^{2}$, $M/M^{\prime}Z(M)\cong\bigoplus_{i\in I}\mathbb{Z}_{p}$, where
$p$ is a prime number, and there exists a maximal subfield $K$ of $D$ such
that $K/F$ is a Galois extension, $\mathrm{Gal}(K/F)\cong M/K^{*}\cap
G\cong\mathbb{Z}_{p}$ and $K^{*}\cap G$ is the $FC$-center and also the
Fitting subgroup of $M$. Furthermore, for any $x\in M\setminus K$ we have
$x^{p}\in F$ and $D=F[M]=\bigoplus_{i=1}^{p}Kx^{i}$.

###### Proof.

Denote by $A$ a maximal abelian normal subgroup of $M$ containing $M^{\prime}$
(we note that such a subgroup exists since $M^{\prime}$ is abelian). Consider
an arbitrary subgroup $N$ of $M$ which properly contains $A$. Since
$M^{\prime}\leq N$, $N\trianglelefteq M$; so the maximality of $M$ in $G$
implies that either $G$ normalizes $F(N)^{*}$ or $F(N)^{*}\cap G\leq M$. If
the second case occurs, then $N\trianglelefteq F(N)^{*}\cap G$, so $N$ is a
metabelian subnormal subgroup of $F(N)^{*}$. By Lemma 3.1, $N$ is abelian,
which contradicts to the maximality of $A$. Hence, $G$ normalizes $F(N)^{*}$
and by Stuth’s theorem we conclude that $F(N)=D$.
Now let $a$ be an element from $M\setminus A$ and assume that $a$ is
transcendental over $F(A)$. Put $T=F(A)^{*}\langle a^{2}\rangle$; since $a$
normalizes $F(A)^{*}$, it is not hard to see that
$F[T]=\bigoplus_{i\in\mathbb{Z}}F(A)a^{2i}$ and $(F[T],F(A),T,T/F(A)^{*})$ is
a crossed product. But $T/F(A)^{*}\cong\langle a^{2}\rangle$ is an infinite
cyclic group and by [[wehfritz], 1.4.3, p. 26], $F[T]$ is an Ore domain. On
the other hand, by what we proved before, we have $F(T)=D$. Therefore, there
exist two elements $s_{1},s_{2}\in F[T]$ such that $a=s_{1}s_{2}^{-1}$.
Writing $s_{1}=\sum_{i=l}^{m}k_{i}a^{2i}$ and
$s_{2}=\sum_{i=l}^{m}k^{\prime}_{i}a^{2i}$, with $k_{i},\ k^{\prime}_{i}\in
F(A)$, for any $l\leq i\leq m$, we have
$\sum_{i=l}^{m}ak^{\prime}_{i}a^{2i}=\sum_{i=l}^{m}k_{i}a^{2i}$. If we set
$l_{i}=ak^{\prime}_{i}a^{-1}$, for any $l\leq i\leq m$, then $l_{i}$’s are
elements of $F(A)$ and we have
$\sum_{i=l}^{m}l_{i}a^{2i+1}=\sum_{i=l}^{m}k^{\prime}_{i}a^{2i}$. This shows
that $a$ is algebraic over $F(A)$, say of degree $n$. Using the fact that $a$
normalizes $F(A)^{*}$, we see that $R=\bigoplus_{i=0}^{n-1}F(A)a^{i}$ is a
domain, which is a finite-dimensional left vector space over $F(A)$. Therefore
$R$ is a division ring and $R=F(A\langle a\rangle)$. By what we proved before
we conclude that $R=D$, hence $[D:F(A)]_{l}<\infty$. So, by Double Centralizer
theorem, $[D:F]<\infty$.
If $A\leq Z(M)$, then, since $M^{\prime}\leq A$, $\langle A,x\rangle$ is an
abelian normal subgroup of $M$ properly containing $A$, for any $x\in
M\setminus A$. But, this contradicts to the maximality of $A$; hence
$A\not\subseteq Z(M)$. Since $[D:F]<\infty$, all elements of $A\setminus Z(M)$
is algebraic over $F$. In view of Corollary 3.6, there exists a subfield $K$
of $D$ such that $K$ and $H=C_{D}(K)$ satisfy $(i)-(iv)$ in Theorem 3.5. So,
$H^{*}\cap G$ is a metabelian subnormal subgroup of $H^{*}$. By Lemma 3.1,
$H^{*}\cap G\subseteq Z(H)=K$. Thus, $(iv)$ implies that $K=H$ is a maximal
subfield of $D$. Since $M$ is metabelian, $M/K^{*}\cap G$ is simple and
metabelian. We conclude that $\mathrm{Gal}(K/F)\cong M/K^{*}\cap
G\cong\mathbb{Z}_{p}$, where $p$ is a prime number and so $[D:F]=p^{2}$. Since
$[M:K^{*}\cap G]=p$ and $D$ is algebraic over $F$, for any $x\in M\setminus
K$, we have $D=F(M)=F[M]=\sum_{i=1}^{p}Kx^{i}$ and, since $[D:K]=p$,
$D=\bigoplus_{i=1}^{p}Kx^{i}$. Suppose $x^{p}\not\in F$. By $F(M)=D$, we have
$Z(M)=M\cap F^{*}$, and it follows $C_{M}(x^{p})\neq M$. On the other hand, we
note that $\langle x,K^{*}\cap G\rangle\leq C_{M}(x^{p})$ and $[M:K^{*}\cap
G]$ is prime, so we get $C_{M}(x^{p})=M$, a contradiction. Thus, $x^{p}\in F$.
Now, by Corollary 2.5, for any $y\in K^{*}\cap M=K^{*}\cap G$, we have
$y^{p}\in M^{\prime}F^{*}$. Therefore $y^{p}\in M^{\prime}(M\cap
F^{*})=M^{\prime}Z(M)$, so $M/M^{\prime}Z(M)$ is an abelian group of exponent
$p$ and by Bear-Prufer’s theorem [22, p. 105],
$M/M^{\prime}Z(M)\cong\bigoplus_{i\in I}\mathbb{Z}_{p}$. This completes the
proof. ∎

In [4], Ebrahimian proved that if $M$ is a nilpotent maximal subgroup of
$D^{*}$, then $M$ is abelian. The following theorem shows that this remains
also true if we replace $D^{*}$ by any its subnormal subgroup.

######  Theorem 3.8.

Let $D$ be a division ring and $G$ a subnormal subgroup of $D^{*}$. If $M$ is
a nilpotent maximal subgroup of $G$, then $M$ is abelian.

###### Proof.

Assume that $M$ is non-abelian. Then, there exists an element $x\in
Z_{2}(M)\setminus Z(M)$. By considering the homomorphism $f:M\to Z(M)$ given
by $f(y)=xyx^{-1}y^{-1}$ we see that $M/C_{M}(x)$ is an abelian group, so
$M^{\prime}\leq C_{M}(x)$. Put $F=Z(D)$; since $x\not\in Z(M)$, we have
$F(M^{\prime})\neq D$. Now we claim that $M^{\prime}$ is abelian. If $G$
normalizes $F(M^{\prime})$, then, by Stuth’s theorem, we have
$F(M^{\prime})\subseteq F$, so $M^{\prime}$ is abelian, as claimed. Now,
suppose that $G$ does not normalize $F(M^{\prime})$. Since $M$ normalizes
$F(M^{\prime})$ and $M$ is maximal in $G$, it follows that $F(M^{\prime})\cap
G\subseteq M$. Thus, $M^{\prime}\trianglelefteq F(M^{\prime})\cap G$ is a
nilpotent subnormal subgroup of $F(M^{\prime})^{*}$ and so, by Lemma 3.1,
$M^{\prime}$ is abelian, as claimed. Therefore $M$ is metabelian. By Theorem
3.7, we conclude that the Fitting subgroup of $M$ is a proper subgroup of $M$,
a contradiction. Thus, $M$ is abelian as it was required to prove. ∎

######  Lemma 3.9.

Let $D$ be a centrally finite division ring with center $F$. Then,
$D^{\prime}\cap F^{*}$ is finite.

###### Proof.

Suppose $[D:F]=n^{2}$. By taking the reduced norm, we obtain $x^{n}=1$ for all
$x\in D^{\prime}\cap F^{*}$. Since $F$ is a field, $D^{\prime}\cap F^{*}$ is
finite. ∎

######  Lemma 3.10.

Let $D$ be a division ring with center $F,G$ a subnormal subgroup of $D^{*}$
and suppose $M$ is a non-abelian maximal subgroup of $G$. If $M^{\prime}$ is
locally finite, then $M/Z(M)$ is locally finite, $M^{\prime}$ is locally
cyclic and the conclusions of Theorem 3.7 follow.

###### Proof.

Suppose $M^{\prime}$ is non-abelian, so we need only to consider the case
${\rm Char}D$=0.
First, we claim that there exists a torsion abelian normal subgroup of $M$
which is not contained in $Z(M)$. By maximality of $M$ in $G$, either
$F(M^{\prime})\cap G\subseteq M$ or $G$ normalizes $F(M^{\prime})$. If
$F(M^{\prime})\cap G\subseteq M$, then $M^{\prime}$ is a locally finite
subnormal subgroup of $F(M^{\prime})^{*}$, so $M^{\prime}$ is abelian by Lemma
3.1, a contradiction. So, $G$ normalizes $F(M^{\prime})$ and by Stuth’s
theorem we have $F[M^{\prime}]=F(M^{\prime})=D$. On the other hand, by [24,
2.5.5, p. 74], there exists a metabelian normal subgroup of $M^{\prime}$ of
finite index $n$. Let $Q=\langle\\{x^{n}|x\in M^{\prime}\\}\rangle$. Then, $Q$
is a metabelian normal subgroup of $M$ and $Q^{\prime}$ is a torsion abelian
normal subgroup of $M$. If $Q^{\prime}$ is not contained in $Z(M)$ then we are
done. Hence, we may assume that $Q^{\prime}\leq Z(M)$, so $Q$ is locally
finite and nilpotent. Thus, by [24, 2.5.2, p. 73], $Q$ contains an abelian
subgroup $B$ such that $[Q:B]<\infty$. Clearly, $C=\cap_{x\in M}xBx^{-1}$ is a
torsion abelian normal subgroup of $M$ and we may assume that $C\leq Z(M)$. On
the other hand, $[Q:B]<\infty$ and $Q\trianglelefteq M$, so it follows that
$Q/C$ has bounded exponent. From definition of $Q$ and $C\leq Z(M)$,
$M^{\prime}/Z(M^{\prime})$ has also bounded exponent. Therefore, by the
classification theorem of locally finite groups in division ring [24, 2.5.9,
p. 75], we conclude that $M^{\prime}$ is abelian-by-finite. Now, let $A$ be an
abelian normal subgroup of $M^{\prime}$ of finite index. Since
$F[M^{\prime}]=D$, $D$ is finite-dimensional over a field $F[A]$. The Double
Centralizer theorem implies that $[D:F]<\infty$ and by Lemma 3.9, $|A\cap
F^{*}|<\infty$. Also by $F[M^{\prime}]=D$, we have $Z(M^{\prime})=F\cap
M^{\prime}$ and hence, $A/A\cap F^{*}$ has bounded exponent, so is $A$. Since
$A$ can be considered as the multiplicative subgroup of a field, we conclude
that $A$ is finite, so is $M^{\prime}$. Hence, $M$ is an $FC$-group and so it
is abelian by Theorem 3.5, a contradiction. Therefore, there exists a torsion
abelian normal subgroup of $M$ which is not contained in $Z(M)$.
Now, by Corollary 3.6, there exists a field $K$ such that $K$ and $H=C_{D}(K)$
satisfy $(i)-(iv)$ in Theorem 3.5. Taking $N=M^{\prime}\cap H^{*}$, we have
$N\trianglelefteq M\cap H^{*}=G\cap H^{*}$, so $N$ is a locally finite
subnormal subgroup of $H^{*}$. By Lemma 3.1, $N=M^{\prime}\cap H^{*}$ is
abelian and so $M^{\prime}\not\subseteq M\cap H^{*}$. If $M/M\cap H^{*}$ is
abelian, then $M^{\prime}\subseteq M\cap H^{*}$, a contradiction. Thus, from
$(iii)$ in Theorem 3.5, $M^{\prime}/N\cong M^{\prime}(M\cap H)/M\cap H=M/M\cap
H$ is a non-abelian finite simple group. Let $\\{x_{1},\ldots,x_{m}\\}$ be a
transversal of $N$ in $M^{\prime}$ and $S=\langle x_{1},\ldots,x_{m}\rangle$ a
finite subgroup of $M^{\prime}$. Since $S$ has a quotient group which is a
finite simple non-abelian group, $S$ is unsolvable. The classification theorem
of finite groups in division ring [24, 2.1.4, p. 46] states that the only
unsolvable finite subgroup of a division ring is $SL(2,5)$, so
$M^{\prime}=S\cong SL(2,5)$ is a finite group. Thus, $M$ is an $FC$-group and
so it is abelian by Theorem 3.5, a contradiction. Therefore, $M^{\prime}$ is
abelian and it can be considered as a torsion multiplicative subgroup of a
field. Hence $M^{\prime}$ is locally cyclic and Theorem 3.7 completes the
proof. ∎

##  4 The existence of noncyclic free subgroups

The question on the existence of noncyclic free subgroups in the
multiplicative group of a division ring was posed by Lichtman in [16]. In [5],
Goncalves proved that if $D$ is centrally finite division ring (i.e. $D$ is
finite dimensional vector space over its center $F$), then $D^{*}$ contains a
noncyclic free group. Later, in [7], Goncalves and Mandel posed the more
general question: is it true that every subnormal subgroup of the
multiplicative group of a division ring contains a noncyclic free subgroup? In
this work, the authors show that this is true in some particular cases. More
exactly, they proved that a subnormal subgroup $G$ of $D^{*}$ contains a
noncyclic free subgroup if $G$ contains some element $x\in D\setminus F$,
which is algebraic over the center $F$ of $D$ such that either (a) ${\rm
Gal}(F(x)/F)\neq 1$ or (b) $x^{p}\in F$, where $p=2$ or $p={\rm char}(F)>0$.
The affirmative answer was also obtained for centrally finite division rings
by Goncalves in his precedent work [6]. Note that the affirmative answer to
the question above would imply some known theorems like the commutativity
theorems of Kaplansky, Jacobson, Hua, Herstein, Stuth,… For more information
we refer to [11]. In this section we shall prove that the question above has
the affirmative answer for locally finite division rings. Recall that a
division ring $D$ with the center $F$ is locally finite if for every finite
subset $S$ of $D$, the division subring $F(S)$ is finite dimensional vector
space over $F$. It can be prove that in a locally finite division ring every
finite subset generates the centrally finite division subring. From this
observation, the following definition seems to be quite natural: a division
ring $D$ is called weakly locally finite if for every finite subset $S$ of
$D$, the division subring of $D$ generated by $S$ is centrally finite. This
notion is introduced firstly in [9]. Clearly, every locally finite is weakly
locally finite. The converse is not true. In [9], using the well-known method
due to Mal’cev and von Neumann, the authors constructed the example of weakly
locally finite division ring which is not even algebraic. In the next two
lemmas we prove some facts for weakly locally finite division rings.

######  Lemma 4.1.

Let $D$ be a weakly locally finite division ring of characteristic $0$ and $G$
be a subgroup of $D^{*}$. If $G$ contains no noncyclic free subgroup, then $G$
is locally solvable -by-finite.

###### Proof.

Take an arbitrary finite subset $S$ of $G$, and let $K$ be a division subring
of $D$ generated by $S$. Then, $K$ is centrally finite , so we have
$m=[K:Z(K)]<\infty$. Therefore, the subgroup $<S>$ generated by $S$ in $G$ can
be considered as a subgroup of ${\rm GL}_{m}(Z(K))$. By [25, Theorem 1], there
exists a solvable subgroup $H$ of $<S>$ of finite index. If $N=H_{G}$, then
$N$ is a solvable normal subgroup of finite index in $<S>$. Hence, $G$ is
locally solvable-by-finite. ∎

######  Lemma 4.2.

Let $D$ be a weakly locally finite division ring and $G$ be a subgroup of
$D^{*}$. If $G$ contains no noncyclic free subgroup, then $G$ is (locally
solvable) -by-(locally finite).

###### Proof.

Firstly, we claim that $G$ is locally solvable-by-locally finite. If ${\rm
char}(D)=0$, then this claim follows from Lemma 4.1. Assume that ${\rm
char}(D)=p>0$. Then, by the same argument as in the proof of Lemma 4.1, for
any finite subset $S$ of $G$, the subgroup $<S>$ generated by $S$ can be
viewed as a linear group over the field of characteristic $p>0$. By [25,
Theorem 2], $<S>$ contains a solvable normal subgroup $H$ such that $<S>/H$ is
locally finite. Hence, $G$ is locally solvable-by-locally finite. Now, by [24,
3.3.9, p. 103], $G$ is (locally solvable) -by-(locally finite). ∎

######  Lemma 4.3.

Let $D$ be a locally finite division ring and G an arbitrary subgroup of
$D^{*}$. Then either $G$ contains a noncyclic free subgroup or $G$ is abelian-
by-locally finite.

###### Proof.

Assume that $G$ does not contain any noncyclic free subgroups. By Lemma 4.2,
$G$ is (locally solvable)-by-(locally finite). The Wehrfritz’s theorem [24,
5.5.1, p. 199] implies that there exists an abelian normal subgroup $A$ of $G$
such that $G/A$ is locally finite. Therefore, $G$ is abelian-by-locally
finite. ∎

From Lemma 3.1 and Lemma 4.3 we get the following result, which shows that
Conjecture 2 in [7] is true for locally finite division rings.

######  Theorem 4.4.

Let $D$ be a locally finite division ring with center $F$ and $G$ a subnormal
subgroup of $D^{*}$. If $G$ is noncentral, then $G$ contains a noncyclic free
subgroup.

The next theorem can be considered as a broad generalization of the main
theorem in [21].

######  Theorem 4.5.

Let $D$ be a locally finite division ring with center $F,G$ a subnormal
subgroup of $D^{*}$ and suppose $M$ is a non-abelian maximal subgroup of $G$.
If $M$ contains no noncyclic free subgroups, then $[D:F]<\infty$, $F(M)=D$ and
there exists a maximal subfield $K$ of $D$ such that $K/F$ is a Galois
extension, $\mathrm{Gal}(K/F)\cong M/K^{*}\cap G$ is a finite simple group and
$K^{*}\cap G$ is the $FC$-center and also the Fitting subgroup of $M$.

###### Proof.

First, we show that $F(M)=D$. If $F(M)\neq D$, then, by Stuth’s theorem,
$F(M)$ is not normalized by $G$. So, by maximality of $M$ in $G$, we have
$F(M)\cap G=M$. Thus, $M$ is a non-abelian subnormal subgroup of $F(M)^{*}$ no
containing a noncyclic free subgroup, which contradicts to Theorem 4.4. So, we
have $F(M)=D$; hence $Z(M)=M\cap F^{*}$. Now, by Lemma 4.3, there exists an
abelian normal subgroup $A$ of $M$ such that $M/A$ is locally finite. If
$A\subseteq Z(M)$, then $M/Z(M)$ is locally finite, so is $M^{\prime}$ and the
result follows from Lemma 3.10. Assume that $A\not\subseteq Z(M)$. By applying
Corollary 3.6, there exists a subfield $K$ of $D$ such that $H=C_{D}(K)$ and
$K$ satisfy $(i)-(iv)$ in Theorem 3.5. Since $H^{*}\cap G$ is a subgroup of
$M$ and also a subnormal subgroup of $H^{*}$, by Theorem 4.4 we have
$H^{*}\cap G\subseteq K$. Now, the condition $(iv)$ in Theorem 3.5 completes
our proof. ∎

##  5 Maximal subgroups of $GL_{1}(D)$

In this section we investigate the similar problem as in Section 3 for the
particular case, when $G=D^{*}$.

######  Theorem 5.1.

Let $D$ be a division ring with center $F$ and $M$ a maximal subgroup of
$D^{*}$, containing a non-central $FC$-element $\alpha$. By setting
$K=F(\alpha^{M})$, the following conditions hold:
(i) $K$ is a field, $[K:F]<\infty$ and $F[M]=D$. (ii) $K^{*}$ is the Hirsch-
Plotkin radical of $M$. (iii) $F^{*}<K^{*}\leq C_{D}(K)^{*}<M<D^{*}$. (iv)
$K/F$ is a Galois extension, $M/C_{D}(K)^{*}\cong\mathrm{Gal}(K/F)$ is a
finite simple group. (v) For any $x\in K,\ |x^{M}|\leq[K:F]$ and for any $y\in
D\setminus K,\ |y^{M}|=|D|$.

###### Proof.

We note that $K$ and $H=C_{D}(K)$ satisfy $(i)-(iii)$ in Theorem 3.5 (for
$G=D^{*}$). As in first paragraph of the proof of Theorem 3.5, we can write
$D=\sum_{i=1}^{t}x_{i}H$, where $x_{1},\ldots,x_{t}\in M$. In this case, we
have $H^{*}\leq M$, and this implies $F[M]=D$. It remains to prove $(ii)$ and
$(v)$. Let $x$ be an element of $K^{*}$. Since $K$ is algebraic over $F$ and
$K^{*}\vartriangleleft M$, $x^{M}\subseteq K$ and the elements of $x^{M}$ have
the same minimal polynomial $f$ over $F$. So
$|x^{M}|\leq\textrm{deg}(f)\leq[K:F]$. For any $y\in D\setminus K$, $C_{H}(y)$
is a proper division subring of $H$ since $y\not\in K=C_{D}(H)$. By [23,
14.2.1, p. 429] and $[D:H]<\infty$ we have
$|y^{M}|=[M:C_{M}(y)]\geq[H^{*}:C_{H}(y)^{*}]=|H|=|D|$. Thus, $(i),(iii),(iv)$
and $(v)$ hold.
Now we prove $(ii)$. Let $A$ be the Hirsch-Plotkin radical of $M$ and suppose
$K^{*}<A$. Then, $B=H^{*}\cap A$ is a locally nilpotent normal subgroup of
$H^{*}$. Hence, by Lemma 3.2, we conclude that $B\subseteq Z(H)=K$ and
consequently, $B=K^{*}$. If $A\subseteq H^{*}$, then $A=B=K^{*}$, a
contradiction. Therefore, $A\not\subseteq H^{*}$. Thus, $AH^{*}/H^{*}$ is a
nontrivial normal subgroup of $M/H^{*}$. Since $M/H^{*}$ is simple,
$M/H^{*}=AH^{*}/H^{*}\cong A/B$. Suppose $R=\sum_{i=1}^{m}y_{i}K$, where
$\\{y_{1},\ldots,y_{m}\\}$ is a transversal of $B$ in $A$. Since $A$
normalizes $K^{*}$ and $B\subseteq K$, $R$ is a division ring and
$[R:K]_{r}\leq m$. Since $R$ is a division ring generated by $A$ and $K$, $M$
normalizes $R$; so by maximality of $M$ in $D^{*}$, it follows that either
$D^{*}$ normalizes $R$ or $R^{*}\leq M$. If the second case occurs, then $A$
is a locally nilpotent normal subgroups of $R^{*}$. By Lemma 3.2, $A$ is
abelian and this contradicts to the fact that $K^{*}$ is the Fitting subgroup
of $M$ (Theorem 3.5 $(ii)$). Thus, $D^{*}$ normalizes $R$, and by Stuth’s
theorem we have $R=D$. Therefore, $[D:K]_{r}\leq
m=|M/H^{*}|=|\mathrm{Gal}(K/F)|=[K:F]$. This implies that $[D:F]=m^{2}$ and
$K=H$ is a maximal subfield of $D$. From $M/H^{*}=AH^{*}/H^{*}$ and $K^{*}<A$,
it follows $M=A$. Since $F[M]=D$, $M$ is a locally nilpotent absolutely
irreducible subgroup of $D^{*}$ and by [24, 5.7.11 p. 215], $M/Z(M)$ is
locally finite. Thus, $K/F$ is a nontrivial radical Galois extension and by
[15, 15.13, p. 258], $F$ is algebraic over a finite field, so is $D$.
Therefore, by Jacobson’s theorem [15, p. 219], $D$ is a field, a
contradiction. Thus, $K^{*}=A$ is the Hirsch-Plotkin radical of $M$. ∎

The proof of the following corollary is similar as the proof of Corollary 3.6.

######  Corollary 5.2.

Let $D$ be a division ring with center $F$ and suppose $M$ is a maximal
subgroup of $D^{*}$. Assume that $M$ contains an abelian normal subgroup $A$
and there exists some element $\alpha\in A\setminus Z(M)$ such that $\alpha$
is algebraic over $F(Z(M))$. Then, $K=F[A]$ satisfies (i) - (v) in Theorem
5.1.

######  Lemma 5.3.

Let $D$ be a division ring with center $F$ and suppose $M$ is a maximal
subgroup of $D^{*}$ such that $C_{D}(M)=F$ and $F[M]^{*}=M$. If $M^{\prime}$
is algebraic over $F$, then $F[M^{\prime}]$ is an algebraic division
$F$-algebra and $F[M^{\prime}]^{*}\trianglelefteq M$.

###### Proof.

First, we claim that if $x\in M$ and $g(t)=(t-r_{n})\cdots(t-r_{1})\in D[t]$
with $r_{n},\ldots,r_{1}\in x^{M}$, then $g(x)\in M\cup\\{0\\}$. Let
$h(t)=(t-r_{n-1})\cdots(t-r_{1})$, by induction we have $h(x)\in
M\cup\\{0\\}$. If $h(x)=0$, then $g(x)=0$ as claimed. If $h(x)\neq 0$, then,
by [15, (16.3), p. 263], we have $g(x)=(x^{h(x)}-r_{n})h(x)$. Take
$r_{n}=x^{m}$ with $m\in M$, we have
$x^{h(x)}-r_{n}=(x^{m^{-1}h(x)}-x)^{m}=([m^{-1}h(x),x]-1)^{m}x^{m}$. On the
other hand, $[m^{-1}h(x),x]-1\in M$ since $M^{\prime}$ is algebraic over $F$
and $F[M]^{*}=M$. Thus, $x^{h(x)}-r_{n}\in M$ and so $g(x)\in M$.
Next, by the same argument as in the proof of Theorem 2.2 and what we proved,
if $x\in M$ is algebraic over $F$ with minimal polynomial $f(t)$ of degree
$n$, then there exists $x_{1},\ldots,x_{n-1}\in x^{M}$ such that

$f(t)=(t-x_{n-1})\cdots(t-x_{1})(t-x)\in D[t].$

Also by the same argument as in the proof of Corollary 2.4, we have $x^{n}\in
M^{\prime}F^{*}$. Thus, if $x,y\in M$ are algebraic over $F$, then
$xM^{\prime}F^{*}$ and $yM^{\prime}F^{*}$ are two elements of
$M/M^{\prime}F^{*}$ of finite order and so $xyM^{\prime}F^{*}$ is of finite
order. Since $M^{\prime}$ is algebraic over $F$, so is $xy$. If $x,y\in M$ are
algebraic over $F$, then, by $F[M]^{*}=M$, $x+y=x(1+x^{-1}y)$ is algebraic
over $F$ and belongs to $M$. Thus, $F[M^{\prime}]$ is an algebraic division
$F$-algebra and $F[M^{\prime}]^{*}\subseteq F[M]^{*}=M$. This completes our
proof. ∎

In the next theorem we get some result as in [1, Theorem 6], but with a weaker
condition. In fact, we replace the condition of algebraicity of $M$ by the
condition of algebraicity of derived subgroup $M^{\prime}$.

######  Theorem 5.4.

Let $D$ be a division ring with center $F$ and $M$ a non-abelian locally
solvable maximal subgroup of $D^{*}$ with $M^{\prime}$ is algebraic over $F$.
Then, $[D:F]=p^{2}$, $M/M^{\prime}F^{*}\cong\bigoplus_{i\in I}\mathbb{Z}_{p}$,
where $p$ is a prime number, and there exists a maximal subfield $K$ of $D$
such that $K/F$ is a Galois extension, $\mathrm{Gal}(K/F)\cong
M/K^{*}\cong\mathbb{Z}_{p}$ and $K^{*}$ is the $FC$-center and also the
Hirsch-Plotkin radical of $M$. Furthermore, for any $x\in M\setminus K^{*}$ we
have $x^{p}\in F,\ M=\bigcup_{i=1}^{p}K^{*}x^{i}$ and
$D=F[M]=\bigoplus_{i=1}^{p}Kx^{i}$.

###### Proof.

If $F(M)\neq D$, then by maximality of $M$, $M\cup\\{0\\}=F(M)$ is a division
ring. By Remark 2 in [4], $M$ is abelian, a contradiction. Hence $F(M)=D$ and
$C_{D}(M)=F$. Suppose that $F[M]\neq D$, so $F[M]^{*}=M$ by maximality of $M$.
By Lemma 5.3, $F[M^{\prime}]$ is a division ring whose multiplicative group is
locally solvable. The Remark 2 in [4] implies that $M^{\prime}$ is abelian, so
by Theorem 3.7, we have $F[M]=D$, a contradiction. Therefore, $M$ is
absolutely irreducible and by Corollary 4 of [27], $M$ is abelian-by-locally
finite. Thus, there exists an abelian normal subgroup $A$ of $M$ such that
$M/A$ is locally finite. We need to show that $D$ is locally finite over $F$.
Let us examine two possible cases.
Case 1: $A\cap M^{\prime}\subseteq F$
We note that $F(M^{\prime})^{*}$ is normalized by $M$. By maximality of $M$,
either $F(M^{\prime})\cap G\subseteq M$ or $G$ normalizes $F(M^{\prime})^{*}$.
In the first case, $M^{\prime}$ is a subnormal subgroup of
$F(M^{\prime})^{*}$. On the other hand, $M$ is abelian-by-locally finite, so
is $M^{\prime}$. By Lemma 3.1, $M^{\prime}$ is abelian; hence, by Theorem 3.7,
$[D:F]<\infty$, as claimed. In the second case, $G$ normalizes
$F(M^{\prime})^{*}$ and by Stuth’s theorem, either $F(M^{\prime})\subseteq F$
or $F(M^{\prime})=D$. If $F(M^{\prime})\subseteq F$, then, by Theorem 3.7,
$[D:F]<\infty$ and we are done. Suppose that $F(M^{\prime})=D$. We know that
$M^{\prime}/M^{\prime}\cap A\cong AM^{\prime}/A\leq M/A$ is locally finite, so
$M^{\prime}/M^{\prime}\cap F$ is locally finite since $M^{\prime}\cap
A\subseteq F$. Therefore, $D=F(M^{\prime})$ is locally finite over $F$.
Case 2: $A\cap M^{\prime}\not\subseteq F$
Since $M^{\prime}$ is algebraic over $F$, there exists an element $x\in A\cap
M^{\prime}\setminus F$ algebraic over $F$. By Corollary 5.2, there exists a
subfield $K_{1}$ of $D$ such that $K_{1}$ satisfies $(i)-(v)$ in Theorem 5.1.
So, $C_{D}(K_{1})^{*}\leq M$ is abelian-by-locally finite and by Lemma 3.1,
$C_{D}(K_{1})^{*}$ is abelian. Now, by Double Centralizer theorem,
$[D:F]<\infty$. Therefore, $D$ is locally finite. Since $M$ is locally
solvable, $M$ contains no noncyclic free subgroups. By Theorem 4.5, there
exists a maximal subfield $K$ of $D$ as in Theorem 4.5. Since $M$ is locally
solvable and $M/K^{*}$ is a finite simple group, we have
$M/K^{*}\cong\mathbb{Z}_{p}$, where $p$ is a prime number. Thus,
$M^{\prime}\leq K^{*}$ and $M$ is metabelian. Now, the proof is complete by
applying of Theorems 3.7 and 5.1. ∎

In [8, Theorem 3.2], it was proved that, if $D$ is a non-commutative division
ring, algebraic over its center, then any locally nilpotent maximal subgroup
of $D^{*}$ is the multiplicative group of some maximal subfield of $D$. Here,
we are now ready to generalize this result by the following theorem.

######  Theorem 5.5.

Let $D$ be a non-commutative division ring with center $F$ and $M$ a locally
nilpotent maximal subgroup of $D^{*}$. If $M^{\prime}$ is algebraic over $F$,
then $M$ is abelian. Consequently, $M$ is the multiplicative group of some
maximal subfield of $D$.

###### Proof.

If $M$ is non-abelian. Using Theorem 5.4, we conclude that the Hirsch-Plotkin
radical of $M$ is a properly subgroup of $M$, this contradicts to the fact
that $M$ is locally nilpotent. Thus $M$ is abelian. The last conclusion is
evident. ∎

## References

  * [1] S. Akbari, R. Ebrahimian, H. Momenaee Kermani, and A. Salehi Golsefidy, Maximal subgroups of $GL_{n}(D)$, J. Algebra 259 (2003), 201-225.
  * [2] S. Akbari and M. Mahdavi-Hezavehi, On the existence of normal maximal subgroups in division rings, J. Pure Appl. Algebra 171 (2002), 123-131.
  * [3] J. D. Dixon, The Structure of Linear Groups, Van Nostrand, 1971.
  * [4] R. Ebrahimian, Nilpotent maximal subgroups of $GL_{n}(D)$, J. Algebra 280 (2004), 244-248.
  * [5] Jairo Z. Goncalves, Free subgroups of units in group rings, Canad. Math. Bull. 27(3) (1984), 309-312.
  * [6] Jairo Z. Goncalves, Free groups in subnormal subgroups and the residual nilpotence of the group of units of group rings, Can. Math. Bull. 27 (1984), 365-370.
  * [7] Jairo Z. Goncalves and Arnando Mandel, Are there free groups in division rings? Israel J. of Math., Vol. 55, No. 1, 1986, 69-80.
  * [8] Bui Xuan Hai and Nguyen Van Thin, On locally nilpotent subgroups of $GL_{1}(D)$, Comm. Algebra 37 (2009), 712-718.
  * [9] Bui Xuan Hai, Mai Hoang Bien, Trinh Thanh Deo, On linear groups over weakly locally finite division rings, Algebra Colloq., in press.
  * [10] R. Hazrat and A.R. Wadsworth, On maximal subgroups of the multiplicative group of a division algebra, J. Algebra 322(7) (2009), 2528-2543.
  * [11] I. N. Herstein, Noncommutative rings, The Carus Math. Monographs, The Math. Soc. of America, 1968.
  * [12] I. N. Herstein, Multiplicative commutators in division ring, Israel Journal of Math. Vol. 31 (1987), No. 2, 180-188.
  * [13] O. H. Kegel, B. A. F. Wehrfritz, Locally finite groups, North Holland, Amsterdam, 1973.
  * [14] T. Keshavarzipour, M. Mahdavi-Hezavehi, On the non-triviality of $G(D)$ and the existence of maximal subgroups of $GL_{1}(D)$, Journal of Algebra 285 (2005), 213-221.
  * [15] T. Y. Lam, A First Course in Noncommutative Rings, GMT 131, Springer, 1991.
  * [16] A. I. Lichtman, On subgroups of the multiplicative group of skew fields, Proc. Amer. Math. Soc., Vol. 63, No. 1, (Mar. 1977), pp. 15-16.
  * [17] A. I. Lichtman, Free subgroups of normal subgroups of the multiplicative group of skew fields, Proc. Amer. Math. Soc., Vol. 71, No. 2 (1978), 174-178.
  * [18] A. I. Lichtman, On normal subgroups of the multiplicative group of skew fields generated by a polycyclic-by-finite group, J. Algebra 78 (1982), 548-577.
  * [19] M. Mahdavi-Hezavehi, S. Akbari-Feyzaabaadi, M. Mehraabaadi, H. Hajie-Abolhassan, On derived groups of division rings II, Comm. Algebra, Volume 23 (1995), Issue 8, 2881 - 2887.
  * [20] M. Mahdavi-Hezavehi, M. G. Mahmudi and S. Yasamin, Finitely generated subnormal subgroups of $GL_{n}(D)$ are central, J. Algebra 225 (2000), 517-521.
  * [21] M. Mahdavi-Hezavehi, Free subgroups in maximal subgroups of $GL_{1}(D)$, J. Algebra 241 (2001), 720-730.
  * [22] Derek J. S. Robinson, A Course in the Theory of Groups, Second Edition, Springer, 1995.
  * [23] W. R. Scott, Group Theory, Dover Publication, INC, 1987.
  * [24] M. Shirvani and B. A. F. Wehrfritz, Skew Linear Groups, Cambridge Univ. Press, Cambridge, 1986.
  * [25] J. Tits, Free subgroups in linear groups, J. Algebra 20 (1972), 250-270.
  * [26] B. A. F. Wehrfritz, Infinite linear groups, Springer, 1973.
  * [27] B. A. F. Wehrfritz, Soluble and locally soluble skew linear groups, Arch. Math. 49 (1987), 379-388.


#  PROSPECTS FOR PHOTON-PHOTON MEASUREMENTS WITH TAGGED PROTONS IN ATLAS

Maciej Trzebiński
The Henryk Niewodniczański Institute of Nuclear Physics Polish Academy of
Sciences
Radzikowskiego 152, 31-342 Cracow, Poland
on behalf of the ATLAS Collaboration

###### Abstract

In 2017, ATLAS has been equipped with a new, dedicated detector system
allowing measurements of forward protons scattered at small angles in
diffractive and electromagnetic processes. These ATLAS Forward Proton
detectors (AFP) can operate during the standard high-luminosity LHC runs and
collect large amounts of integrated luminosity. This gives a possibility to
study rare interactions, in particular, the two-photon processes. The physics
programme includes measurements of photon-photon interactions present in the
Standard Model, as well as using searches for new physics. In this paper, the
AFP detectors and physics goals are briefly presented.

##  1 Introduction

††* Copyright 2019 CERN for the benefit of the ATLAS Collaboration. CC-BY-4.0
license.

In the majority of events of photon-photon and photon-proton scatterings at
the LHC one or both outgoing protons stay intact. Since photon is a colourless
object, such an exchange results in a presence of the rapidity gap between the
centrally produced system and scattered protons. Thus, such events are of
diffractive nature.

Diffractive processes are an important part of the physics programme at hadron
colliders. This is also true for ATLAS[1], where a large community works on
both phenomenological and experimental aspects of diffraction. In such events,
a rapidity gap111Rapidity gap is a space in rapidity devoid of particles.
between the centrally produced system and scattered protons is present. Due to
the exchange of a colourless object, a photon (in case of electromagnetic
interaction) or Pomeron (strong force), one or both outgoing protons may stay
intact.

##  2 Detection Techniques

The diffractive production may be recognized by the search for a rapidity gap
in the forward direction or by the measurement of scattered protons. The first
method is historically a standard one for the diffractive pattern recognition.
It uses the usual detector infrastructure as trackers and forward
calorimeters. Unfortunately, the rapidity gap may be destroyed by e.g.
particles coming from the pile-up – parallel, independent collisions happening
in the same bunch crossing. In addition, the gap may be outside the acceptance
of a detector. In the second method, protons are directly measured. This
solves the problems of gap recognition in the very forward region and a
presence of pile-up. However, since protons are scattered at small angles (few
hundreds microradians), additional devices called forward detectors are needed
to be installed.

###  2.1 ATLAS Forward Proton Detectors

ATLAS is equipped with two sets of forward proton detectors: ALFA [2, 3] and
AFP [4]. ALFA (Absolute Luminosity For ATLAS) detectors are designed to
measure the properties of the elastic cross-section, soft diffraction and low-
mass exclusive production. These topics are not in the scope of this report –
readers interested in details should see e.g. Ref. [5, 6] (properties of the
elastic scattering measured by ATLAS at $\sqrt{s}=7$ and $8$ TeV), Ref. [7]
(general overview; soft diffraction), Ref. [8] (exclusive pion production) or
Ref. [9] (diffractive bremsstrahlung).

AFP (ATLAS Forward Proton) consists of four detector stations placed
symmetrically with respect to the ATLAS Interaction Point at 205 m and 217 m.
In each AFP station there is a Roman Pot device allowing the units to move
horizontally. Detectors located on the ATLAS C side are inserted into the beam
1 whereas the ones on the A side into the beam 2. The scheme of AFP detectors
is shown in Figure 1.

Figure 1: Scheme of the ATLAS Forward Proton detectors.

Each AFP station consists of four Silicon Trackers (SiT), which provide
precise position measurements. The purpose of the AFP tracking system is to
measure points along the trajectory of protons that were deflected during a
proton-proton interaction. The readout chip was chosen to be FE-I4, which was
originally designed for the IBL project [10]. There are four such chips
installed in each station. The active area covered by the tracking detector is
approximately $16\times 20$ mm2 with a pixel size of $50\times 250$ $\mu$m2.
Detectors are tilted by 14 degrees. The resolution of a single plane was
measured to be about 6 $\mu$m in $x$ and about 30 $\mu$m in $y$ [11]. By
having two detectors on each side of the Interaction Point (IP) one can
measure not only the position of the proton with respect to the beam, but also
its elevation angle. These are connected to the proton kinematics at the
interaction point – i.e. by measuring the proton properties in the AFP one can
unfold its initial four-momentum [12].

Far stations host also the Time of Flight (ToF) detectors. The timing
detectors measure the time of arrival of each proton, providing a trigger
signal and allowing background reduction through the difference in proton
time-of-flight measured on each side of the interaction point. The vertex
calculated from ToF difference on both AFP sides can be compared to the
primary interaction vertex. The resolution is expected to be between 20 and 30
ps [11].

###  2.2 LHC Optics

Between the AFP stations and ATLAS interaction point several LHC elements are
placed. They have a significant impact on the proton trajectory and will
influence its position in the AFP stations. These elements are:

  * •

two dipole magnets (D1-D2) used for beam separation (bending),

  * •

five quadrupole magnets (Q1-Q5) used for beam focusing,

  * •

two collimators (TCL4, TCL5) used for magnet protection.

Settings of the LHC magnets are called optics and come from the requirements
of the experiments in terms of luminosity and of the LHC machine protection.
These settings may differ between the LHC fills. Due to the optics settings,
the forward proton trajectories between the IP and AFP detectors are not
straight lines. A typical situation ($\beta^{*}=0.4$ m optics) for the high-
luminosity ATLAS data taking is shown in Figure 2. Black rectangles represent
dipole and quadrupole magnets, blue lines – collimators and red lines – AFP
and ALFA stations. Assuming the proton transverse momentum equals zero,
protons are bent accordingly to the energy lost during the collision:
$\xi=\frac{E_{beam}-E_{proton}}{E_{beam}}$. As one can see, protons with very
small energy loss are too close to the beam to be detected. With increasing
energy loss, trajectories are further away from the beam and can be detected
by AFP. However, if the energy loss is too big, forward protons will be
filtered by collimators and will not reach the AFP station. The acceptance for
a typical low-$\beta^{*}$ optics covers $0.025<\xi<0.1$ [13], which
corresponds to the proton energy loss of $160<E_{proton}<650$ GeV for
$\sqrt{s}=13$ TeV.

Figure 2: Energy dependence of the proton trajectory in $(x,s)$ plane for
$\sqrt{s}=13$ TeV, $\beta^{*}=0.4$ m LHC optics in vicinity of ATLAS
Interaction Point (IP1). Protons were generated at $(0,0,0)$ with transverse
momentum $p_{\mathrm{T}}=0$. The crossing angle in horizontal plane was set to
185 $\mu$rad. For details see Ref. [13].

##  3 Photon Physics with Forward Proton Measurement

Photoproduction physics has so far been studied mainly in the electron
accelerators. However, a high-energy bremsstrahlung from the proton at the LHC
is a plentiful source of photons. Photoproduction processes can be studied
using proton tagging.

###  3.1 Two Photon Processes: $\gamma\gamma\to\mu\mu$

A di-muon system can be produced in the exclusive mode: $p+p\to
p\gamma^{*}\gamma^{*}p\to p\mu^{+}\mu^{-}p$, see Fig. 3. Such measurement was
done by ATLAS without AFP [14]. The used data sample consisted of exclusive
events and large irreducible background which was mainly coming from
dissociated222Events in which one or both protons was destroyed due to
interactions with other particles from the system. Such phenomena are
described by a gap survival probability. events. Information about the
presence of scattered protons should allow a significant reduction of this
background.

Figure 3: Feynman diagrams for the exclusive di-muon photo-production.

Due to the acceptance of AFP, double-tagged events have too small cross-
section to be observed. However, a semi-exclusive (single tag) measurement
should be possible assuming 40 fb-1 of collected data, minimal muon transverse
momentum of 10 GeV and AFP positioned at about 2 mm from the beam. As was
discussed in Ref. [4], such measurement can be used for the AFP detector
alignment and optics calibration.

###  3.2 Anomalous Gauge Couplings

Measurement of $W$ and $Z$ boson pair production via the exchange of two
photons (see left panel of Fig. 4) allows to perform a stringent test of the
electroweak symmetry breaking [15]. Standard Model predicts the existence of
$\gamma\gamma WW$ quartic couplings while there is no $\gamma\gamma ZZ$
coupling. As was shown in Refs [16] and [17], collecting 30 – 300 fb-1 of data
with the ATLAS detector and using protons measured in AFP should result in a
gain in the sensitivity of about two orders of magnitude over a standard ATLAS
analysis.

Figure 4: Diagrams of anomalous gauge coupling (left) and magnetic monopole
(right) production.

###  3.3 New Physics Searches

Proton tagging may also serve as a powerful technique for the new physics
searches as the backgrounds can be significantly reduced by the kinematic
constraints coming from the AFP proton measurements. The general idea of
background reduction was presented in Ref. [18, 19, 20] on a basis of the
exclusive jet measurement.

Proton tagging technique might be also used for the invisible object searches.
As an example, the case of magnetic monopoles produced by the photon exchange
can be considered. From a diagram (see right panel of Fig. 4) one can conclude
that, even if the centrally produced system escapes detection (or is not
measurable) in ATLAS, one can measure scattered protons in AFP. In general,
any production of new objects (with mass up to 2 TeV) via photon or gluon
exchanges should be possible to be observed.

##  4 Summary

Since 2017 ATLAS is equipped with a full set of the AFP detectors, which
collected data with a proton tag on both sides during the special and standard
LHC runs. Even more data is planned to be collected during the LHC Run 3.
Besides QCD measurements (rapidity gap survival, Pomeron structure, etc.),
photon-induced processes can be measured. These include single-tagged
exclusive muons ($pp\to p\mu^{+}\mu^{-}p$) and anomalous gauge couplings ($W$,
$Z$ and photon pairs). For the latter processes the use of the AFP detectors
provides a significant gain in sensitivity as compared to the measurement
based on the data from the central ATLAS detector. On top of that one can try
to search for any production of new objects produced via photon or gluon
exchanges (magnetic monopoles, invisible particles, …). In such searches,
forward proton measurements can be used for a background reduction.

## References

  * 1 .  ATLAS Collaboration, The ATLAS Experiment at the CERN Large Hadron Collider, JINST 3 (2008) S08003.
  * 2 .  ATLAS Collaboration, ATLAS Forward Detectors for Measurement of Elastic Scattering and Luminosity, CERN-LHCC-2008-004, ATLAS-TDR-18, https://cds.cern.ch/record/1095847.
  * 3 .  S. Abdel Khalek et al., The ALFA Roman Pot Detectors of ATLAS, JINST 11 (2016) P11013.
  * 4 .  ATLAS Collaboration, Technical Design Report for the ATLAS Forward Proton Detector, CERN-LHCC-2015-009, ATLAS-TDR-024, https://cds.cern.ch/record/2017378.
  * 5 .  ATLAS Collaboration, Measurement of the total cross section from elastic scattering in pp collisions at $\sqrt{s}$ = 7 TeV with the ATLAS detector, Nuclear Physics B 889 (2014) 486.
  * 6 .  ATLAS Collaboration, Measurement of the total cross section from elastic scattering in pp collisions at $\sqrt{s}$ = 8 TeV with the ATLAS detector, Phys. Lett. B 761 (2016) 158.
  * 7 .  K. Akiba et al., LHC Forward Physics, SLAC-PUB-16364, DESY 15-167.
  * 8 .  R. Staszewski, P. Lebiedowicz, M. Trzebinski, J. Chwastowski, A. Szczurek, Exclusive $\pi^{+}\pi^{-}$ Production at the LHC with Forward Proton Tagging, Acta Physica Polonica B 42 (2011) 1861.
  * 9 .  J. J. Chwastowski, S. Czekierda, R. Staszewski, M. Trzebinski, Diffractive Bremsstrahlung at High-$\beta^{*}$ LHC Case Study, Eur. Phys. J. C 77 (2017) 216.
  * 10 .  ATLAS Collaboration, ATLAS Insertable B-Layer Technical Design Report, CERN-LHCC-2010-013, ATLAS-TDR-19.
  * 11 .  J. Lange et al., Beam tests of an integrated prototype of the ATLAS Forward Proton detector, JINST 11 (2016) P09005.
  * 12 .  M. Trzebinski, R. Staszewski, J. Chwastowski, LHC High Beta* Runs: Transport and Unfolding Methods, ISRN High Energy Physics 2012 (2012) 491460.
  * 13 .  M. Trzebinski, Machine Optics Studies for the LHC Measurements, Proc. SPIE 9290, Photonics Applications in Astronomy, Communications, Industry, and High-Energy Physics Experiments 2014, 929026.
  * 14 .  ATLAS Collaboration, Exclusive dilepton production at 7 TeV, Physics Letters B 749 (2015) 242-261.
  * 15 .  P. J. Dervan, A. Signer, W. J. Stirling, A. Werthenbach, Anomalous triple and quartic gauge boson couplings, J. Phys. G 26 (2000) 607.
  * 16 .  E. Chapon, O. Kepka, C. Royon, Anomalous quartic $WW\gamma\gamma$, $ZZ\gamma\gamma$, and trilinear $WW\gamma$ couplings in two-photon processes at high luminosity at the LHC, Phys. Rev. D 81 (2010) 074003.
  * 17 .  O. Kepka, C. Royon, Anomalous $WW\gamma$ coupling in photon-induced processes using forward detectors at the LHC, Phys. Rev. D 78 (2008) 073005.
  * 18 .  ATLAS Collaboration, Exclusive Jet Production with Forward Proton Tagging Feasibility Studies for the AFP Project, ATL-PHYS-PUB-2015-003, https://cds.cern.ch/record/1993686.
  * 19 .  M. Trzebinski, R. Staszewski, J. Chwastowski, On the Possibility of Measuring the Single-tagged Exclusive Jets at the LHC, Eur. Phys. J. C 75 (2015) 320.
  * 20 .  M. Trzebinski, Exclusive Jet Measurement in Special LHC Runs – Feasibility Studies, Acta Phys. Pol. B 47 (2016) 1745.