# Enumeration of left braces with additive group $C_{4}\times C_{4}\times
C_{4}$

A. Ballester-Bolinches  Departament de Matemàtiques, Universitat de València,
Dr. Moliner, 50, 46100 Burjassot, València, Spain<EMAIL_ADDRESS>,
R. Esteban-Romero  Departament de Matemàtiques, Universitat de València, Dr.
Moliner, 50, 46100 Burjassot, València, Spain<EMAIL_ADDRESS>and  V.
Pérez-Calabuig  Departament de Matemàtiques, Universitat de València, Dr.
Moliner, 50, 46100 Burjassot, València, Spain<EMAIL_ADDRESS>
###### Abstract.

We show that the number of isomorphism classes of left braces of order $64$
with additive group isomorphic to $C_{4}\times C_{4}\times C_{4}$ is
$1\,515\,429$.

###### Key words and phrases:

Skew left brace, left brace, regular subgroup, holomorph

###### 2020 Mathematics Subject Classification:

Primary 16T25; Secondary 81R50, 20-08

##  1\. Introduction

The notion of _skew left brace_ was introduced by Guarnieri and Vendramin in
[9]. A _skew left brace_ is a triple $(B,{+},{\cdot})$ where $B$ is a set and
${+}$, ${\cdot}$ are two binary operations on $B$ such that $(B,{+})$ and
$(B,{\cdot})$ are groups and that are related by the distributive-like law
$a(b+c)=ab-a+ac$. As it is common in the theory of skew left braces, we omit
the sign $\cdot$ and we write $ab$ instead of $a\cdot b$ and we use $-a$ to
denote the inverse of $a$ in the group $(B,{+})$; the expression $a-b$ means
$a+(-b)$. When, in addition, $(B,{+})$ is an abelian group, then we speak of a
_left brace_ , a notion introduced by Rump in his seminal paper [12].

One of the most natural problems in the theory of skew left braces is the
determination of skew left braces of a given finite order. Guarnieri and
Vendramin present in [9, Algorithm 5.1] an algorithm to enumerate all skew
left braces with a given finite additive group $A$. They presented in [9] the
numbers of isomorphism classes of left braces of order $n$ for $n\leq 120$
except for $n\in\\{32,64,81,96\\}$ obtained with their implementation of this
algorithm in Magma [6]. In fact, they claim in their paper: “With current
computational resources, we were not able to compute the number of non-
isomorphic left braces of orders $32$, $64$, $81$ and $96$.” This computation
appears open as [9, Problem 6.1]. Vendramin posed in [13, Problem 2.13] the
problem of constructing all left braces of order $32$, for which he presented
some partial results on [13, Table 2.3]. He also wrote as a comment to this
problem: “The number of (skew) left braces of size $64$, $96$ or $128$ seems
to be extremely large and our computational methods are not strong enough to
construct them all.”

Bardakov, Neshchadim, and Yadav presented in [5, Algorithm 2.4] a modification
of [9, Algorithm 5.1] for the computation of finite skew left braces with a
given additive group. They were able to enumerate the isomorphisms classes of
skew left braces of orders $32$ and $81$, as well as the left braces of order
$96$. With respect to the isomorphism classes of left braces of order $64$,
they were able to enumerate them in [5, Table 6] for all isomorphism classes
of abelian groups of order $64$, except for the cases of additive group
isomorphic to $C_{4}\times C_{4}\times C_{4}$ (SmallGroup(64, 55) in the
notation of GAP [8]) and to $C_{2}\times C_{2}\times C_{4}\times C_{4}$
(SmallGroup(64, 192)).

In this paper we obtain the isomorphism classes of left braces with additive
group isomorphic to $C_{4}\times C_{4}\times C_{4}$.

######  Theorem 1.1.

There are $1\,515\,429$ isomorphism classes of left braces of order $64$ whose
additive group is isomorphic to $C_{4}\times C_{4}\times C_{4}$.

Table 1 summarises these isomorphism classes by the isomorphism class of the
multiplicative group. For instance, the first two entries of the first row of
the table ($4$, $42$) mean that there are two isomorphism classes of braces
with additive group $C_{4}\times C_{4}\times C_{4}$ and multiplicative group
isomorphic to SmallGroup(64, 4).

Table 1. Number of isomorphism classes of left braces with additive group $C_{4}\times C_{4}\times C_{4}$ by multiplicative group id | # | id | # | id | # | id | # | id | #
---|---|---|---|---|---|---|---|---|---
4 | 42 | 75 | 43 525 | 119 | 67 | 165 | 81 | 221 | 26 932
5 | 40 | 76 | 14 422 | 120 | 66 | 166 | 130 | 222 | 5 404
6 | 4 | 77 | 28 837 | 121 | 67 | 167 | 1 | 223 | 32 308
7 | 22 | 78 | 43 304 | 122 | 66 | 168 | 47 | 224 | 2 763
8 | 74 | 79 | 43 180 | 123 | 1 | 169 | 46 | 225 | 2 731
9 | 64 | 80 | 14 420 | 124 | 40 | 170 | 47 | 226 | 21 671
10 | 16 | 81 | 43 123 | 125 | 40 | 172 | 46 | 227 | 64 744
13 | 2 | 82 | 4 118 | 128 | 63 | 173 | 34 | 228 | 21 644
14 | 16 | 83 | 32 | 129 | 135 | 175 | 34 | 229 | 14 412
17 | 50 | 84 | 33 | 130 | 163 | 176 | 48 | 230 | 3 618
18 | 82 | 85 | 42 | 131 | 92 | 177 | 34 | 231 | 10 802
19 | 12 | 86 | 32 | 132 | 164 | 178 | 82 | 232 | 43 179
20 | 156 | 87 | 102 | 133 | 224 | 179 | 1 | 233 | 21 576
21 | 36 | 88 | 93 | 134 | 287 | 181 | 1 | 234 | 43 092
22 | 24 | 89 | 112 | 135 | 98 | 182 | 1 | 235 | 10 786
23 | 406 | 90 | 947 | 136 | 326 | 192 | 976 | 236 | 10 794
24 | 48 | 91 | 266 | 137 | 130 | 193 | 3 453 | 237 | 10 790
25 | 106 | 92 | 149 | 138 | 485 | 194 | 2 826 | 240 | 10 828
32 | 294 | 93 | 58 | 139 | 515 | 195 | 6 945 | 241 | 21 686
33 | 283 | 94 | 164 | 140 | 1 | 196 | 16 440 | 242 | 3 664
34 | 133 | 95 | 57 | 141 | 42 | 197 | 3 624 | 243 | 21 516
35 | 160 | 96 | 101 | 142 | 73 | 198 | 5 468 | 244 | 10 820
36 | 2 | 97 | 109 | 143 | 114 | 199 | 6 987 | 246 | 13
37 | 10 | 98 | 135 | 144 | 73 | 200 | 489 | 247 | 55
55 | 567 | 99 | 65 | 145 | 138 | 201 | 5 430 | 248 | 56
56 | 3 757 | 100 | 105 | 146 | 131 | 202 | 7 632 | 249 | 66
57 | 3 640 | 101 | 316 | 147 | 35 | 203 | 19 248 | 250 | 12
58 | 21 838 | 102 | 280 | 148 | 96 | 204 | 13 610 | 251 | 43
59 | 21 628 | 103 | 39 | 149 | 131 | 205 | 16 629 | 252 | 31
60 | 3 908 | 104 | 42 | 150 | 35 | 206 | 27 099 | 253 | 120
61 | 21 812 | 105 | 32 | 151 | 96 | 207 | 8 277 | 254 | 147
62 | 11 052 | 106 | 17 | 152 | 64 | 208 | 5 407 | 255 | 203
63 | 10 850 | 107 | 17 | 153 | 16 | 209 | 9 052 | 256 | 246
64 | 7 193 | 108 | 21 | 154 | 48 | 210 | 32 312 | 257 | 52
65 | 3 682 | 109 | 67 | 155 | 2 | 211 | 1 919 | 258 | 144
66 | 43 617 | 110 | 24 | 156 | 43 | 212 | 2 763 | 259 | 92
67 | 43 927 | 111 | 24 | 157 | 2 | 213 | 13 608 | 260 | 193
68 | 86 219 | 112 | 52 | 158 | 43 | 214 | 2 758 | 261 | 1 011
69 | 87 259 | 113 | 54 | 159 | 1 | 215 | 19 242 | 262 | 173
70 | 43 183 | 114 | 46 | 160 | 66 | 216 | 19 092 | 263 | 2 052
71 | 21 837 | 115 | 68 | 161 | 52 | 217 | 8 253 | 264 | 1 921
72 | 21 725 | 116 | 140 | 162 | 52 | 218 | 13 543 | 265 | 489
73 | 14 585 | 117 | 68 | 163 | 69 | 219 | 53 836 | 266 | 503
74 | 14 420 | 118 | 1 | 164 | 81 | 220 | 32 333 | 267 | 10

The study of the left braces with additive group isomorphic to $C_{2}\times
C_{2}\times C_{4}\times C_{4}$ has been the object of a later research in [4],
after the submission of this paper. That article has been accepted for
publication while writing the revised version for this paper. We must indicate
that the techniques used in this paper to classify the left braces with
additive group $C_{4}\times C_{4}\times C_{4}$ up to isomorphism are not
enough to solve the corresponding problem for the additive group $C_{2}\times
C_{2}\times C_{4}\times C_{4}$ due mainly to the high number of intermediate
subgroups needed in the algorithm and we have had to use new ideas that are
described with detail in [4]. The main results of that paper are the following
ones.

######  Theorem 1.2 ([4, Theorem 1.1]).

The number of isomorphism classes of left braces of order $64$ with additive
group isomorphic to $C_{2}\times C_{2}\times C_{4}\times C_{4}$ is
$10\,326\,821$.

######  Corollary 1.3 ([4, Corollary 1.2]).

The number of isomorphism classes of left braces of order $64$ is
$15\,095\,601$.

##  2\. The holomorph of a group

Most of the results in this section can be considered as folklore. We present
them here for the sake of completeness.

Let $(G,{+})$ be a group. The _holomorph_ of $G$ is

$\operatorname{Hol}(G,{+})=\\{(g,\alpha)\mid g\in
G,\alpha\in\operatorname{Aut}(G)\\}$

with the operation given by

$(g,\alpha)(h,\beta)=(g+\alpha(h),\alpha\circ\beta).$

The identity element of $\operatorname{Hol}(G,{+})$ is $(0,1)$ and the inverse
of the element $(g,\alpha)\in\operatorname{Hol}(G,{+})$ is

$(g,\alpha)^{-1}=(-\alpha^{-1}(g),\alpha^{-1}).$

The subgroup $A=\\{(0,\alpha)\mid\alpha\in\operatorname{Aut}(G,{+})\\}$ is
isomorphic to $\operatorname{Aut}(G,{+})$. We identify $A$ with
$\operatorname{Aut}(G,{+})$.

The group $\operatorname{Hol}(G,{+})$ acts on $(G,{+})$ by means of

$(g,\alpha)*h=g+\alpha(h),\qquad g,h\in G.$

Note that when $(G,{+})$ is the additive group of a vector space, then
$\operatorname{Hol}(G,{+})$ can be identified with the affine group on the
vector space $G$ and this corresponds to the natural action of the affine
group on $G$.

Furthermore, let us show that this action is faithful: If
$(g,\alpha)*h=(k,\beta)*h$ for all $h\in G$, then $g+\alpha(h)=k+\beta(h)$ for
all $h\in G$. In particular, taking $h=0$, we obtain that $g=k$. Consequently,
$\alpha(h)=\beta(h)$ for all $h\in G$ and so $\alpha=\beta$. Hence, this
action is faithful. Therefore, we can identify $\operatorname{Hol}(G,{+})$
with a subgroup of the group $\Sigma_{G}$ of all permutations of the set $G$.

Given $g\in G$, let $\tau_{g}\colon G\longrightarrow G$ be given by
$\tau_{g}(h)=g+h$, the _left translation_ of $G$ defined by $g$. It is clear
that $T=\\{\tau_{g}\mid g\in G\\}$ is a subgroup of $\Sigma_{G}$ isomorphic to
$G$. Furthermore, we can identify $\tau_{g}$ with
$(g,1)\in\operatorname{Hol}(G,{+})$. The proof of the following proposition
can be found, for instance, in [11, Application 1.5.2].

######  Proposition 2.1.

The normaliser in $\Sigma_{G}$ of $T$ coincides with the holomorph of
$(G,{+})$.

We now present some characterisations of regular subgroups that are useful for
our purposes.

First of all, note that a subgroup $H$ of
$\operatorname{Hol}(G,{+})\leq\Sigma_{G}$ is regular if, and only if, it acts
transitively on $G$ and the stabiliser $H_{g}$ of each element $g\in G$ is
trivial. In other words, given $h$, $k\in G$, there exists a unique
$(g,\alpha)\in H$ such that $(g,\alpha)*k=h$. Suppose that $H$ is a regular
subgroup of $\operatorname{Hol}(G,{+})$. Given $h\in G$, there exists a unique
$(g,\alpha)\in H$ such that $(g,\alpha)*0=h$. Consequently, $g+\alpha(0)=g=h$.
Suppose that $(g,\alpha)$, $(g,\beta)\in H$. Since
$g=(g,\alpha)*0=(g,\beta)*0=g$, the regularity of the action of $H$ on $G$
shows that $\alpha=\beta$. Hence, $\alpha$ is uniquely determined by $g\in G$,
let us call $\alpha=\lambda_{g}$. Consequently $H=\\{(g,\lambda_{g})\mid g\in
G\\}$, where the map $\lambda\colon G\longrightarrow G$,
$\lambda(g)=\lambda_{g}$ depends on $H$. Furthermore, since the action is
transitive, the “projection” of $H$ on the $G$-component of
$\operatorname{Hol}(G,{+})$ is surjective.

Note that $(0,1)*0=0$, therefore $\lambda_{0}=1$ and so
$H\cap\operatorname{Aut}(G,{+})=\\{(0,1)\\}$.

######  Remark 2.2.

Other authors have considered the regularity of a subgroup $H$ of
$\operatorname{Hol}(G)$ in the following equivalent way: given $k\in G$, there
exists a unique $(h,\beta)\in H$ such that $(h,\beta)*k=0$, that is, for every
$k\in G$ there exists a unique $(h,\beta)\in H$ such that $h+\beta(k)=0$. We
have preferred the opposite point of view because we obtain more easily the
expression $H=\\{(g,\lambda_{g})\mid g\in G\\}$ for a regular subgroup $H$ of
$\operatorname{Hol}(G)$ (cf. [9, Lemma 4.1]).

In the following proposition, we assume that $(G,{+})$ is a finite group.

######  Proposition 2.3.

Let $(G,{+})$ be a finite group and let $H$ be a subgroup of
$\operatorname{Hol}(G,{+})$. Let us denote by $\pi_{G}$ the “projection” of
$\operatorname{Hol}(G,{+})$ on $G$. Then $\lvert
H\rvert=\lvert\pi_{G}(H)\rvert\lvert H\cap\operatorname{Aut}(G,{+})\rvert$.

###### Proof.

The orbit of $0\in G$ with respect to the action of $\operatorname{Hol}(G)$ on
$G$ is $\\{(g,\alpha)*0\mid(g,\alpha)\in H\\}=\pi_{G}(H)$ and the stabiliser
of $0$ is $\\{(g,\alpha)\in H\mid(g,\alpha)*0=0\\}=\\{(g,\alpha)\in H\mid
g=0\\}=\operatorname{Aut}(G)\cap H$. The result follows as an application of
the orbit-stabiliser theorem. ∎

Proposition 2.3 is useful in the finite case to discard subgroups whose
subgroups cannot contain regular subgroups because the “projection” to $G$ is
not surjective in the computation of all regular subgroups of the holomorph of
a given finite group $(G,{+})$. We can do it by means of the following result.

######  Proposition 2.4.

Let $(G,{+})$ be a finite group and let $H$ be a subgroup of
$\operatorname{Hol}(G,{+})$. The restriction of the “projection” $\pi_{G}$ of
$\operatorname{Hol}(G,{+})$ to $H$ is surjective if, and only if, $\lvert
H\rvert=\lvert G\rvert\lvert H\cap\operatorname{Aut}(G,{+})\lvert$.

Another consequence of Proposition 2.3 is the following characterisation of
regular subgroups of $\operatorname{Hol}(G,{+})$ for a finite group $(G,{+})$.

######  Proposition 2.5 (cf. [1]).

Let $(G,{+})$ be a finite group. Every two of the following three statements
about a subgroup $H$ of $\operatorname{Hol}(G,{+})$ imply the other one.

  1. (1)

$\lvert H\rvert=\lvert G\rvert$.

  2. (2)

$H\cap\operatorname{Aut}(G,{+})=\\{(0,1)\\}$.

  3. (3)

The restriction to $H$ of the “projection” $\pi_{G}$ of
$\operatorname{Hol}(G,{+})$ on $G$ is surjective.

Moreover, a subgroup $H$ of $\operatorname{Hol}(G,{+})$ satisfying two of the
three previous properties (and so the other one) is regular.

###### Proof.

The fact that every two of the three statements imply the other one is an
immediate consequence of Proposition 2.4. As in the proof of Proposition 2.3,
$\pi_{G}(H)$ is the orbit of $0$ under the action of $H$ and
$H\cap\operatorname{Aut}(G)$ is the stabiliser of $0$. If $H$ satisfies all
these properties, then the orbit of $0$ is $G$ and its stabiliser is trivial,
that is, $H$ is regular. ∎

######  Proposition 2.6.

Let $H$ be a regular subgroup of $\operatorname{Hol}(G)$, say
$H=\\{(g,\lambda_{g})\mid g\in G\\}$. Then, given $g$, $k\in G$,
$\lambda_{g+\lambda_{g}(k)}=\lambda_{g}\circ\lambda_{k}$ and
$\lambda_{g}^{-1}=\lambda_{\lambda_{g}^{-1}(-g)}$

###### Proof.

Note that
$(g,\lambda_{g})(k,\lambda_{k})=(g+\lambda_{g}(k),\lambda_{g}\circ\lambda_{k})=(g+\lambda_{g}(k),\lambda_{g+\lambda_{g}(k)})\in
H$. When we apply this to $k=\lambda_{g}^{-1}(-g)$, we obtain that
$1=\lambda_{0}=\lambda_{g}\circ\lambda_{\lambda_{g}^{-1}(-g)}$. The result
follows. ∎

The following fact is mentioned in [5] and used to improve the algorithms to
obtain all skew left braces with a given additive group.

######  Proposition 2.7.

Let $H$ be a regular subgroup of $\operatorname{Hol}(G,{+})$ with $(G,{+})$ a
group and let $(g,\alpha)\in\operatorname{Hol}(G,{+})$. Then
$(g,\alpha)H(g,\alpha)^{-1}$ is again a regular subgroup of
$\operatorname{Hol}(G,{+})$. Furthermore, there exists
$\beta\in\operatorname{Aut}(G,{+})$ such that
$(g,\alpha)H(g,\alpha)^{-1}=(0,\beta)H(0,\beta)^{-1}$.

##  3\. Regular subgroups and skew left braces

We present now the result that allows us to construct all skew left braces
with a given additive group $(G,{+})$. All these results are well known (see,
for instance, [9, Section 4]) and we present them here for completeness.

######  Proposition 3.1.

Let $(B,{+},{\cdot})$ be a skew left brace. Given $a\in B$, let
$\lambda_{a}\colon B\longrightarrow B$ be the lambda map given by
$\lambda_{a}(b)=-a+ab$. Then $H=\\{(a,\lambda_{a})\mid a\in B\\}$ is a regular
subgroup of $\operatorname{Hol}(B,{+})$.

###### Proof.

It is well known that $\lambda_{a}\in\operatorname{Aut}(B,{+})$ for all $a\in
A$. Since

$(a,\lambda_{a})(b,\lambda_{b})=(a+\lambda_{a}(b),\lambda_{a}\lambda_{b})=(a+\lambda_{a}(b),\lambda_{a+\lambda_{a}(b)})\in
H$

by Proposition 2.6 and, by the same result,
$\lambda_{a}^{-1}=\lambda_{\lambda_{a}^{-1}(-a)}$ and so
$(a,\lambda_{a})^{-1}=(\lambda_{a}^{-1}(-a),\lambda_{a}^{-1})=(\lambda_{a}^{-1}(-a),\lambda_{\lambda_{a}^{-1}(-a)})\in
H$, $H$ is a subgroup of $\operatorname{Hol}(B,{+})$. Given $a\in G$, there
exists a unique element $(g,\alpha)$ in $H$ such that $(g,\alpha)*0=a$, namely
$(g,\alpha)=(a,\lambda_{a})$. Therefore, $H$ is regular. ∎

The proof of the following proposition can be found in [9, Theorem 4.2].

######  Proposition 3.2.

Given a regular subgroup $H$ of $\operatorname{Hol}(G,{+})$, with $(G,{+})$ a
group, then $H$ admits a structure of skew left brace whose additive group is
isomorphic to $(G,{+})$.

The following result combines Lemma 2.1 and Theorem 2.2 of [5].

######  Proposition 3.3.

Two regular subgroups $H_{1}$ and $H_{2}$ of $\operatorname{Hol}(G,{+})$
induce isomorphic skew left braces if, and only if, they are conjugate by an
element of $\operatorname{Aut}(G,{+})$.

Note that, by Proposition 2.7, the condition of Proposition 3.3 can be
replaced by conjugation in $\operatorname{Hol}(G,{+})$.

##  4\. Computational challenges

By Section 3, the problem of determining the skew left braces with additive
group isomorphic to $G=C_{4}\times C_{4}\times C_{4}$ up to isomorphism is
reduced to determining the conjugacy classes of regular subgroups of
$\operatorname{Hol}(G)$. Furthermore, since $G$ is a $2$-group, every regular
subgroup $H$ of $\operatorname{Hol}(G)$ has order $\lvert H\rvert=\lvert
G\rvert=64$ and so a conjugate of $H$ is contained in a fixed Sylow
$2$-subgroup of $\operatorname{Hol}(G)$. Hence it is enough to determine all
regular subgroups of a Sylow $2$-subgroup of $\operatorname{Hol}(G)$.

Hulpke [10] has developed an algorithm to determine the lattice of subgroups
of a finite soluble group $S$. This algorithm is implemented by means of the
function SubgroupsSolvableGroup of GAP [8]. We summarise this algorithm as
follows:

  1. (1)

We compute a normal series $S\trianglerighteq
N_{1}\trianglerighteq\dots\trianglerighteq N_{r}=1$ with elementary abelian
factors.

  2. (2)

We construct by induction the subgroups of $S/N_{i+1}$ from the subgroups of
$S/N_{i}$. Without loss of generality, we assume that $N_{i+1}=1$, $N=N_{i}$,
and we know the subgroups of $S/N$. We have the following possibilities for a
subgroup $U$ of $S$:

    1. (a)

$U$ contains $N$ and thus $U$ is the full preimage of a subgroup of $S/N$
under the natural epimorphism;

    2. (b)

$U$ is contained in $N$ and so $U$ is a subspace of the vector space $N$, or

    3. (c)

$B:=U\cap N$ is a proper subgroup of $N$ and $A=NU$ is a subgroup of $G$ that
contains properly $N$.

The subgroups of the first type are simply the preimages of the subgroups of
$S/N$, that have been computed by induction. The subgroups of the second type
are the subspaces of the vector space $N$. Hence it is enough to consider the
third case. In the third case, $B\trianglelefteq U$ and $B\trianglelefteq N$.
Therefore $B\trianglelefteq NU=A$ and
$A\leq\operatorname{N}_{S}(A)\cap\operatorname{N}_{S}(B)$. It follows that
$U/B$ can be computed as a complement of $N/B$ in $A/B$.

We cannot apply this algorithm directly to $S=\operatorname{Hol}(G)$ since
$\operatorname{Hol}(G)$ is not soluble, but we can apply it to a Sylow
$2$-subgroup of $\operatorname{Hol}(G)$. The implementation of
SubgroupsSolvableGroup in GAP includes the possibility of adding restrictions
like ExactSizeConsiderFunction to avoid the computation of subgroups that do
not lead to subgroups of the specified order. This is useful since regular
subgroups of $\operatorname{Hol}(G)$ have the same order as $G$. We also note
that regular subgroups of $\operatorname{Hol}(G)$ must have a surjective
“projection” onto $G$ and so we can add the restriction of Proposition 2.4 to
discard all subgroups leading only to non-regular subgroups.

In the GAP implementation of SubgroupsSolvableGroup, the list of all conjugacy
classes of subgroups of $G/N_{i}$ (layer $i$) and all computed conjugacy
classes of $G/N_{i+1}$ (layer $i+1$) are stored at each layer. However, only
one group of the layer $i$ is needed at each step and the groups obtained for
the layer $i+1$ will not be used until advancing to the next layer.
Furthermore, they can be a lot of subgroups and they can use a large amount of
memory, that could eventually exhaust the physical memory. Our approach is to
replace saving these subgroups to the memory by saving them to a hard disk at
the obvious drawback of speed. Furthermore, in the event of a power loss, we
could restart the computation at the exact point it was stopped. We have also
modified the algorithm to use space on a hard disk instead of the RAM.

We note that the GAP function SubgroupsSolvableGroup, when it is applied with
some restrictions like ExactSizeConsiderFunction, returns all subgroups
satisfying these restrictions, but it might return other subgroups. Our
implementation includes a final check to remove these eventually extra
subgroups.

##  5\. Our computations

The implementation of this modified algorithm for the computation of the
regular subgroups of a Sylow $2$-subgroup of $\operatorname{Hol}(C_{4}\times
C_{4}\times C_{4})$ produced $31\,367\,678$ conjugacy classes. Of course, some
of these classes can have representatives that are not conjugate in the Sylow
$2$-subgroup, but can be conjugate in $\operatorname{Hol}(C_{4}\times
C_{4}\times C_{4})$. Consequently, the next natural step is to classify their
representatives by conjugation in $\operatorname{Hol}(C_{4}\times C_{4}\times
C_{4})$. Since we were expecting many regular subgroups to be compared by
conjugation in $\operatorname{Hol}(C_{4}\times C_{4}\times C_{4})$, in the
last step of the algorithm we classify the regular groups by their isomorphism
class, the isomorphism class of the kernel of the action of the brace on the
additive group (the set of all elements of the regular subgroup that stabilise
all elements of $C_{4}\times C_{4}\times C_{4}$, that coincides with the
centraliser of the normal subgroup $C_{4}\times C_{4}\times C_{4}$ in the
holomorph as a semidirect product with respect to this action) and the
isomorphism class of the quotient by this normal subgroup. We have obtained an
overall number of $1\,442$ equivalence classes.

Our idea is to reduce the checking of conjugation to each of these $1\,442$
equivalence classes. The comparisons of different equivalence classes can be
performed in parallel by using different processors. Some of these equivalence
classes turn out to be small, for example, $74$ of them have only one element
and $1\,055$ have at most $100$ elements. In all these classes the comparison
by conjugation is fast. However, $63$ equivalence classes have more than
$100\,000$ subgroups, $19$ equivalence classes have more than $500\,000$
subgroups, $13$ equivalence classes have more than $800\,000$ subgroups, and
the $4$ largest equivalence classes have more than $1\,000\,000$ subgroups.
The largest one has $1\,782\,312$ subgroups.

We have decided to refine the $63$ largest equivalence classes by means of the
length of the conjugacy class in $\operatorname{Hol}(C_{4}\times C_{4}\times
C_{4})$. This refinement applied to all equivalence classes gives a total
number of $2\,353$ equivalence classes. In some cases, this has allowed us to
obtain some small numbers of subgroups that can be easily compared by
conjugation, but for the largest conjugacy class lengths the computations were
still slow. The execution of these comparisons on a computer with an Intel
processor i7-11700 that allows the execution of $16$ parallel tasks and $32$
Gb of RAM running GNU/Linux during a couple of months made us guess that we
would need more than two years to perform the task.

At that time we applied for the use of the scientific supercomputer _Lluís
Vives_ to the Computer Service of the Universitat de València (see [7]). We
thank the Computer Service for granting an immediate access to this machine
and for their help installing GAP and solving our doubts. On this machine, we
were able to complete the computations in less than two months by running
several comparisons in parallel. Our implementation of parallelism in this
setting has consisted of running several instances of GAP that select from a
list of regular subgroups a unique representative of each conjugacy class in
$\operatorname{Hol}(C_{4}\times C_{4}\times C_{4})$ or select from two lists
of regular subgroups for which two elements in the same list are not conjugate
in $\operatorname{Hol}(C_{4}\times C_{4}\times C_{4})$ a list of the subgroups
of both lists that contain no pairs of conjugate subgroups. We have not used
any particular computer package to run GAP in parallel mode, as our setting
has been enough for our purposes. The total number of conjugacy classes of
regular subgroups of $\operatorname{Hol}(C_{4}\times C_{4}\times C_{4})$ we
have found is $1\,515\,429$. Table 1 summarises the numbers of conjugacy
classes by the first invariant we consider, the isomorphism class of the
multiplicative group of the resulting left brace.

## Acknowledgements

These results are part of the R+D+i project supported by the Grant
PGC2018-095140-B-I00, funded by MCIN/AEI/10.13039/501100011033 and by “ERDF A
way of making Europe.” We thank the anonymous referees for their comments that
have allowed us to improve the proofs of some results and the presentation of
the paper. We also thank the Computer Service of the Universitat de València
for allowing us to use the scientific supercomputer _Lluís Vives_ and for
their kind support in setting up the system for the computation and solving
our questions.

## Data availability

The complete list of left braces or order $64$ is available for use in GAP [8]
on https://github.com/RamonEstebanRomero/braces64 [3]. It also includes the
left braces with additive group $C_{2}\times C_{2}\times C_{4}\times C_{4}$
computed by the authors in [4]. The storage of these left braces follows the
ideas of [2] of representing them as triply factorised groups. This repository
also includes some GAP functions to use these left braces with the help of the
YangBaxter package [14] for GAP.

## References

  * [1] E. Acri and M. Bonatto, _Skew braces of order $p^{2}q$ I: Abelian type_, Alg. Colloq. 29 (2020), no. 2, 297–320.
  * [2] A. Ballester-Bolinches and R. Esteban-Romero, _Triply factorised groups and the structure of skew left braces_ , Commun. Math. Stat. 10 (2022), 353–370.
  * [3] A. Ballester-Bolinches, R. Esteban-Romero, and V. Pérez-Calabuig, _Left braces of order $64$_, WWW https://doi.org/10.5281/zenodo.7406729, December 2022, DOI:10.5281/zenodo.7406729.
  * [4] by same author, _Enumeration of left braces with additive group $C_{2}\times C_{2}\times C_{4}\times C_{4}$_, Ric. Mat. (2023), https://doi.org/10.1007/s11587-023-00772-2.
  * [5] V. G. Bardakov, M. V. Neshchadim, and M. K. Yadav, _Computing skew left braces of small orders_ , Int. J. Algebra Comput. 30 (2020), no. 4, 839–851.
  * [6] W. Bosma, J. J. Cannon, C. Fieker, and A. Steel, _Handbook of magma functions_, 2.19 ed., 2016.
  * [7] Computer Service, Universitat de València, _Lluisvives v2 user’s guide_ , Universitat de València, 2020, https://www.uv.es/calcul/Manuales/LluisVives2_user_manual.pdf (visited 8th April, 2022).
  * [8] The GAP Group, _GAP – Groups, Algorithms, and Programming, Version 4.11.1_ , 2021.
  * [9] L. Guarnieri and L. Vendramin, _Skew-braces and the Yang-Baxter equation_ , Math. Comp. 86 (2017), no. 307, 2519–2534. MR 3647970
  * [10] A. Hulpke, _Computing subgroups invariant under a set of automorphsms_ , J. Symbolic Comput. 27 (1999), no. 4, 415–427.
  * [11] A. Kerber, _Applied finite group actions_ , 2nd ed., Algorithms and Combinatorics, vol. 19, Springer-Verlag, Berlin, Heidelberg, New York, 1999.
  * [12] W. Rump, _Braces, radical rings, and the quantum Yang-Baxter equation_ , J. Algebra 307 (2007), 153–170. MR 2278047
  * [13] L. Vendramin, _Problems on skew left braces_ , Adv. Group Theory Appl. 7 (2019), 15–37.
  * [14] L. Vendramin and O. Konovalov, _YangBaxter: combinatorial solutions for the Yang-Baxter equation_ , August 2022, version 0.10.1, https://gap-packages.github.io/YangBaxter/.


# Coupling between multiple antennas through a plasma cylinder

L. Chang1,2, L. P. Zhang1,2, X. G. Yuan1,2, Y. J. Chang1,2, J. H. Zhang1,2, X.
Yang1, Y. Wang1, H. S. Zhou1,2, and G. N. Luo1,2,3 1Institute of Plasma
Physics, HFIPS, Chinese Academy of Sciences, Hefei $230031$, China  2Science
Island Branch of Graduate, University of Science and Technology of China,
Hefei $230026$, China  3School of Nuclear Science and Technology, University
of Science and Technology of China, Hefei $230026$, China

###### Abstract

The coupling physics between multiple antennas separated axially along a
plasma cylinder is investigated. Experiments are carried out on a recently
built device: Physics ANd Thruster oriented HElicon Research (PANTHER), with
an upgrade of second-stage antennas. Mutual induction currents are measured in
detail. It is found that the existence of plasma column going through these
antennas increase the coupling effects among them significantly. Theoretical
analyses from the perspectives of transformer and magnetic permeability and
moment confirm the reasonability of this phenomenon. This work is of
particular interest for electrodeless plasma source or thruster which employs
multiple antennas for ionisation and acceleration.

Keywords: mutual induction, electrodeless acceleration, helicon discharge

PACS: 52.25.Mq, 52.40.Fd, 52.50.Qt, 52.75.Di, 52.80.Pi

##  1 Introduction

The scheme of antenna wrapping around a quartz tube has been employed widely
for inductively coupling and helicon discharges[1, 2, 3]. This electrodeless
feature can lead to long life and high efficiency, due to the absence of
electrode erosion and plasma sheath, respectively. To further increase the
plasma density and temperature, together with particle energy, multiple
antennas were conceived and implemented, such as VASIMR (VAriable Specific
Impulse Magnetoplasma Rocket)[4], HEAT (Helicon Electrodeless Acceleration
Thruster)[5] and other experiments[6, 7, 8, 9]. However, the mutual coupling
between these coaxial antennas, which are separated axially with a certain
distance, through the same plasma cylinder was rarely studied. This coupling
can result in significant magnitude of voltage and current on neighbouring
antennas and, according to the Faraday’s law and the conservation of magnetic
flux, they are generally in the opposite direction to those originally
provided by their power supplies. With regard to plasma generation, this
surely weakens the ionisation rate and heating efficiency. Thereby, it is of
great importance to investigate the coupling physics between these antennas,
and guide experiments to either reduce the coupling effect or turn the
detrimental coupling into beneficial usage. Here, we explore the coupling
physics on a recently built device which employs three antennas around a
helicon plasma cylinder, and through analyses from the perspectives of
transformer and magnetic permeability.

##  2 Experimental setup

We conduct the experiments on PANTHER (Physics ANd Thruster oriented HElicon
Research)[10], which was built recently and produced blue-core helicon plasma,
with an upgrade of second-stage antennas. Figure 1 shows a schematic. It has
four main parts: vacuum chamber, power supplies, solenoid coils and
diagnostics. The vacuum chamber comprises an outer stainless steel cylinder in
length of $2$ m and diameter of $0.5$ m, which is equally and hermetically
separated in the axial direction to form source region and heat & diffusion
region, respectively, and an inner quartz tube in length of $1.5$ m and
diameter of $0.1$ m. The vacuum environment surrounding the antenna in the
source region, which represents a special feature of PANTHER for space-related
research, is maintained by rotary (AVT-$16.7$ L/s) and turbo-molecular
(CBVAC-$1600$ L/s) pumps, while the low-pressure environment inside the quartz
tube and the heat & diffusion region is pumped by rotary (KYKY-$14$ L/s) and
turbo-molecular (KYKY-$1600$ L/s) pumps. The pressure is monitored by vacuum
gauges (Leybord PTR 90N). The working gas (helium throughout the paper) is fed
into the quartz tube from the upstream of source region, and controlled via a
series of mass-flow controllers in range of $0\sim 200$ sccm. The background
pressure in level of $10^{-5}$ Pa can be achieved, whereas the working
pressure is around $10^{-1}$ Pa. The power supplies consist of $13.56$
MHz-$1000$ W and $2\sim 30$ MHz-$500$ W. They are connected through matching
networks to a half-turn helical antenna ($A_{0}$) for plasma generation in the
source region and two isolated loop antennas ($A_{1}$ and $A_{2}$) with phase
difference of $\pi/2$ for heating in the diffusion region, respectively.
Please note that for the initial study of coupling physics in the present
work, only one loop ($A_{2}$) is powered, while the other loop ($A_{1}$) is
shortened by a resistance ($R_{\ast}=1~{}\Omega$) to measure the purely
induced voltage and current; moreover, the driving frequency is fixed to
$13.56$ MHz for both antennas, to simplify the wave activity and focus on the
coupling process. There are four magnetic coils surrounding the vacuum
chamber, which are powered and cooled separately to provide equilibrium
magnetic field with adjustable profile. The magnetic field strength is up to
$0.18$ T. The diagnostics include radio-frequency compensated Langmuir Probe
(LP) to measure the electron density and temperature, two voltage-current (VI)
probes to monitor the output from matching network to antenna, and Optical
Emission Spectrometer (OES) to characterise the intensity and width of light
emission which can then be used to infer the plasma parameters.

Figure 1: Schematic of experimental setup: upgrade from PANTHER (Physics ANd
Thruster oriented HElicon Research)[10] with second-stage antennas.

##  3 Data & Analyses

###  3.1 Coupling without plasma

We first measured the inductance of individual antennas for typical driving
frequency range through a network analyser (KEYSIGHT-$4$ GHz), and carried it
out in vacuum environment and without discharge. Figure 2 shows the results.
It can be seen that the inductance grows monotonically with driving frequency,
as expected, and the inductance of $A_{0}$ is slightly bigger than those of
$A_{1}$ and $A_{2}$, especially in high frequency range, due to larger
diameter, more turns, and longer axial extension. For the particular frequency
of $13.56$ MHz employed in following sections, the inductance becomes
$L_{A_{0}}=4.278$ $\mu$H, $L_{A_{1}}=4.083$ $\mu$H and $L_{A_{2}}=4.102$
$\mu$H, respectively.

Figure 2: Inductance of individual antennas for typical driving frequency
range. Red dashed line labels the particular frequency of $13.56$ MHz employed
in following sections.

Then we switched on the power supply on $A_{2}$ while keeping the vacuum and
no-discharge environment, and monitored the induced currents on neighbouring
antennas. Figure 3 presents the variation of antenna current with input power
exerted on $A_{2}$. We can see that the antenna currents all increase with
input power, as expected, and the induced currents on $A_{1}$ and $A_{0}$ stay
largely in the same phase of that on $A_{2}$, which is illustrated more
clearly by the normalised results shown in the inset. The slight jump around
$P_{A_{2}}=20$ W may be correlated to mode transition from isolated two-
antenna interactions to integrated three-antenna interactions when the power
exceeds a certain level. Indeed, the coupling between these antennas is
eventually a three-body interaction problem, for example, the induced current
on $A_{0}$ results from coupling with both $A_{2}$ and $A_{1}$, leading to
interference feature displayed by the inset. Please note that the ratios
between the antenna currents do not vary much and on average they are
$I_{A_{0}}/I_{A_{1}}\approx 0.073$, $I_{A_{0}}/I_{A_{2}}\approx 0.047$ and
$I_{A_{1}}/I_{A_{2}}\approx 0.699$, respectively.

Figure 3: Variation of antenna current on input power without plasma
discharge. Inset gives normalised results, showing the same phase of current
on $A_{0}$, $A_{1}$ and $A_{2}$.

To explore the underlying physics, we analyse the induced current from the
perspective of transformer. Figure 4 shows a schematic of magnetic field lines
generated by $A_{2}$. We can see that these antennas do not share the same
magnetic flux, due to spacing and leakage.

Figure 4: Schematic of magnetic field lines generated by $A_{2}$.

We assume that the magnetic flux enclosed by each antenna is $\Psi_{0}$ for
$A_{0}$, $\Psi_{1}$ for $A_{1}$ and $\Psi_{2}$ for $A_{2}$. Since they are
driven by the same power supply with fixed frequency of $f_{\ast}=13.56$ MHz,
they all vary in form of $\exp(i\omega_{\ast}t)$ with $\omega_{\ast}=2\pi
f_{\ast}$. According to the Faraday’s law, the induced voltage $U$ is
determined by time-varying magnetic field flux, i. e.

$U=\oint_{l}\mathbf{E}\cdot
d\mathbf{l}=-\int_{s}\frac{\partial\mathbf{B}}{\partial t}\cdot d\mathbf{s}.$
(1)

Here, $\mathbf{E}$ and $\mathbf{B}$ are time-varying electric and magnetic
fields, respectively, and $l$ and $s$ are the path and area of integrations.
Referring to transformer, we write the equations:

$U_{0}=R_{\ast}I_{0}=-n_{0}\frac{d\Psi_{0}}{dt}=-i\omega_{\ast}n_{0}\Psi_{0},$
(2)
$U_{1}=R_{\ast}I_{1}=-n_{1}\frac{d\Psi_{1}}{dt}=-i\omega_{\ast}n_{1}\Psi_{1},$
(3)
$U_{2}=R_{\ast}I_{2}=-n_{2}\frac{d\Psi_{2}}{dt}=-i\omega_{\ast}n_{2}\Psi_{2},$
(4)

with $n$ the number of coils for each antenna. Therefore, the ratios of
antenna current equals to the ratios of induced magnetic flux ($n\Psi$),
namely

$I_{0}/I_{1}/I_{2}=n_{0}\Psi_{0}/n_{1}\Psi_{1}/n_{2}\Psi_{2}.$ (5)

For the loop antennas of $A_{1}$ and $A_{2}$, we have $n_{1}=n_{2}=1$; while
for the half-turn helical antenna of $A_{0}$, we may write $n_{0}\approx 2.5$.
It can be seen that although $A_{1}$ is very close to $A_{2}$, it only
captures $69.9\%$ of the magnetic flux generated by $A_{2}$; while for
$A_{0}$, the percentage is much lower ($4.7\%/2.5$) because of much longer
distance.

###  3.2 Coupling with plasma

We show above that the coupling between antennas immersed in vacuum
environment can be described simply by the Faraday’s equation. This situation,
however, becomes complicated when there is a plasma cylinder going through
them, as shown in Fig. 1. To reveal the underlying physics, we pumped in
helium with background pressure of $P_{B}=0.042$ Pa, and applied external
magnetic field of strength $B_{0}=0.08$ T. Figure 5 shows the variation of
antenna current with input power exerted on $A_{0}$. Please note that for all
the cases of $P_{A_{2}}=0$ W throughout the paper, the antenna $A_{2}$ is
shortened by a resistance ($R_{\ast}=1~{}\Omega$). This also applies to
$A_{1}$ for which $P_{A_{1}}=0$ W applies.

Figure 5: Variation of antenna current on input power with plasma discharge.
Inset gives normalised results, showing the same phase of current on $A_{0}$,
$A_{1}$ and $A_{2}$.

We find that the current increases with input power for all antennas under the
same phase, consistent with Fig. 3. The ratios between them are independent of
input power and they are $I_{A_{1}}/I_{A_{0}}\approx 0.150$,
$I_{A_{2}}/I_{A_{0}}\approx 0.146$ and $I_{A_{2}}/I_{A_{1}}\approx 0.973$,
respectively. Compared with Fig. 3, one can see that the ratios are much
higher, indicating better coupling between antennas with the existence of
plasma. Since the induced power origins from $A_{0}$ and the distance between
$A_{1}$ and $A_{0}$ is shorter than that between $A_{2}$ and $A_{0}$, it is
reasonable to have $I_{A_{2}}/I_{A_{1}}<1$. Again, we draw the magnetic flux
surrounding these antennas (Fig. 6), and analyse these ratios referring to
transformer but with the inclusion of plasma. As pointed in early
references[11, 12], a stabilised plasma can be paramagnetic, and could
increase the magnetic permeability and confine the magnetic flux, behaving
similarly to the magnetic core for a transformer. This surely enhances the
coupling effect among antennas and thereby the ratios of induced current.
Please note that although plasma cannot be treated as normal magnetic
material[13], the gyrating motions of charged particles do provide additional
magnetic moment, in form of $\mu_{m}=mv_{\perp}^{2}/2B$, which leads to the
increased effective magnetic permeability ($\mu_{e}=\mu_{0}+\mu_{m}$).

Figure 6: Schematic of magnetic field lines generated by $A_{0}$.

Then we switched on the power supply for two antennas, i. e. $A_{0}$ and
$A_{2}$, and compared with one-antenna discharge. Experiments were conducted
for unmagnetised and magnetised cases, respectively. Figure 7 shows the
variation of antenna current with input power on $A_{0}$ without external
magnetic field ($B_{0}=0$ T). The averaged ratios are: (a)
$I_{A_{0}}/I_{A_{1}}\approx 16.249$, $I_{A_{0}}/I_{A_{2}}\approx 9.055$ and
$I_{A_{1}}/I_{A_{2}}\approx 0.763$; (b) $I_{A_{0}}/I_{A_{1}}\approx 4.682$,
$I_{A_{0}}/I_{A_{2}}\approx 2.959$ and $I_{A_{1}}/I_{A_{2}}\approx 0.689$. It
can be seen that the addition of input power on $A_{2}$ slightly lowers the
current on $A_{0}$, which is consistent with the diamagnetic induction between
these two antennas. While the current on $A_{2}$ increases as expected, the
current on $A_{1}$ also increases. This is because the coupling between
$A_{1}$ and $A_{2}$ is stronger than that between $A_{1}$ and $A_{0}$, due to
much shorter distance. Please note that for the case of $P_{a_{2}}=0$ W, the
ratio of $I_{A_{1}}/I_{A_{2}}$ is ideally bigger than unity if simply
considering the distance; however, the mutual induction transits from two-body
to three-body problems when the power becomes higher, making the induction
more complicated. This also applies to the case of $P_{A_{0}}=0$ W shown in
Fig. 8, namely the ratio of $I_{A_{1}}/I_{A_{0}}$ which also involves the
number of antenna coils ($n_{0}/n_{1}\approx 2.5$).

$\begin{array}[]{ll}(a)&(b)\\\
\includegraphics[width=195.12767pt,angle={0}]{figure_7a.eps}&\includegraphics[width=195.12767pt,angle={0}]{figure_7b.eps}\end{array}$

Figure 7: Variation of antenna current with input power without external
magnetic field ($B_{0}=0$ T): (a) $P_{a_{2}}=0$ W, (b) $P_{a_{2}}=20$ W.

For the magnetised case of $B_{0}=0.12$ T, the variation of antenna current
with input power on $A_{2}$ is given by Fig. 8. The averaged ratios are: (a)
$I_{A_{0}}/I_{A_{1}}\approx 3.236$, $I_{A_{0}}/I_{A_{2}}\approx 0.723$ and
$I_{A_{1}}/I_{A_{2}}\approx 0.186$; (b) $I_{A_{0}}/I_{A_{1}}\approx 8.878$,
$I_{A_{0}}/I_{A_{2}}\approx 2.650$ and $I_{A_{1}}/I_{A_{2}}\approx 0.304$.
Similar to Fig. 7, the addition of input power on $A_{0}$ not only increases
the current magnitude of $A_{0}$, as expected, but also increases the induced
current on the unpowered $A_{1}$. However, the addition of external magnetic
field does not show clear trend, which will be confirmed by the parameter
studies in Fig. 10(a). To understand the effects of additional power, we
attribute them to the formed plasma of higher density and temperature, which
increase the number and energy of charged particles, respectively. This yields
larger effective magnetic moment ($\sum{\mu_{m}=\sum{mv_{\perp}^{2}/2B}}$) and
thereby stronger coupling.

$\begin{array}[]{ll}(a)&(b)\\\
\includegraphics[width=195.12767pt,angle={0}]{figure_8a.eps}&\includegraphics[width=195.12767pt,angle={0}]{figure_8b.eps}\end{array}$

Figure 8: Variation of antenna current with input power with external magnetic
field ($B_{0}=0.12$ T): (a) $P_{A_{0}}=0$ W, (b) $P_{A_{0}}=500$ W.

The absolute magnitude of antenna current is eventually determined by the
three-body inductions, as shown in Fig. 9; the induced current on $A_{1}$ is a
result from both $A_{0}$ and $A_{2}$, although the mutual induction between
$A_{0}$ and $A_{2}$ is less effected by the unpowered $A_{1}$.

Figure 9: Schematic of magnetic field lines generated by $A_{2}$ and $A_{0}$.

Finally, we studied the effects of magnetic field and background pressure on
antenna coupling. The applied input power is $P_{A_{0}}=500$ W and
$P_{A_{2}}=0$ W. Figure 10 shows the measurements of antenna current. The
averaged ratios are: (a) $I_{A_{0}}/I_{A_{1}}\approx 7.180$,
$I_{A_{0}}/I_{A_{2}}\approx 6.738$ and $I_{A_{1}}/I_{A_{2}}\approx 0.934$; (b)
$I_{A_{0}}/I_{A_{1}}\approx 8.399$, $I_{A_{0}}/I_{A_{2}}\approx 7.744$ and
$I_{A_{1}}/I_{A_{2}}\approx 0.921$. We can see that, especially from the
normalised results, the current magnitude drops first and then increases to a
stable value when the field strength is increased, minimising around
$B_{0}=0.04$ T; while as the pressure level is enhanced, the current magnitude
drops gradually with oscillations, peaking around $P_{B}=0.7$ Pa for the
pressure range employed. Assuming the microcosmic electron spin that
determines the magnetic permeability remains unchanged, the macroscopic
effects of field and pressure take place only through the magnetic moment of
plasma, i. e. $\mu_{m}=mv_{\perp}^{2}/2B$, either directly (field) or
indirectly (pressure) through collisions.

$\begin{array}[]{ll}(a)&(b)\\\
\includegraphics[width=195.12767pt,angle={0}]{figure_10a.eps}&\includegraphics[width=195.12767pt,angle={0}]{figure_10b.eps}\end{array}$

Figure 10: Variation of antenna current with: (a) external magnetic field
($P_{B}=0.02$ Pa), and (b) background pressure ($B_{0}=0.08$ T) for
$P_{A_{0}}=500$ W and $P_{A_{2}}=0$ W.

Figure 11 displays the discharge images for typical field strengths and
pressure levels. One can see that the plasma is more confined near axis for
higher field and lower pressure. Detailed calculation of the paramagnetic or
diamagnetic property of cylindrical plasma requires three-dimensional profiles
of plasma density and temperature, which are missing at the moment. We thereby
leave it as a separate study and will present in the future.

Figure 11: Discharge images for varied external magnetic field ($a_{1}$:
$0.02$ T, $a_{2}$: $0.04$ T, $a_{3}$: $0.14$ T) and background pressure
($b_{1}$: $0.1$ Pa, $b_{2}$: $0.5$ Pa, $b_{3}$: $0.8$ Pa).

##  4 Conclusion

To reveal the mutual induction between co-axial antennas wrapping around a
plasma cylinder, measurements are carried out for their induced currents for
both unmagnetised and magnetised cases. We find that the induced currents
evolute in the same phase, and their magnitudes are determined eventually by
the multi-antenna inductions through Faraday’s law. The separation distance
between antennas and the existence of plasma column effect these inductions
remarkably, via the leakage of magnetic flux and the enhanced magnetic
permeability respectively. The additional effect of external magnetic field
behaves through the ionisation process, plume confinement and magnetic moment,
whereas the background pressure effects the ionisation and magnetic moment
largely through collisions. Analyses referring to transformer provide
intuitive understanding of this coupled system. In this respect, the plasma
column acts as a magnetic core for a transformer, enhancing the coupling
effect by limiting the path of magnetic flux. The magnetic property of plasma
column, however, relies on three-dimensional profiles of density and
temperature, and is left for detailed calculation in the future.

This work is supported by the Chinese Academy of Sciences “$100$ Talent”
Program (B), the Science Foundation of Institute of Plasma Physics (No.
DSJJ-$2020$-$07$), and the Shanghai Engineering Research Center of Space
Engine (No. $17$DZ$2280800$).

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