cyclic (possibly trivial). In the context of one-relator groups, Brodski[ĭ]{}’s results imply that if the Magnus subsets $S$ and $T$ are disjoint, then $F(S) \cap F(T)$, $gF(S)g^{-1}\cap F(S)$ and $gF(S)g^{-1}\cap F(T)$ are cyclic. Finally Newman [@N] showed that in one-relator groups with torsion, Magnus subgroups are malnormal, ie if $M=F(S)$, where $S$ is a Magnus subset and $g \notin M$, then $gMg^{-1}\cap M$ is trivial

In addition, in [@C], we also showed that, by extending a version of Newman’s argument, one can easily prove that if the one-relator group $G$ has torsion, ie when the relator is a proper power, then, for any two Magnus subgroups $M=F(S)$ and $N=F(T)$ and any $g \in G$, $M \cap N$ is not exceptional and either $gMg^{-1}\cap N$ is trivial or $g\in NM$. Moreover, Newman’s approach – using the so-called Spelling Lemma – also yields, in the torsion case, an algorithm to determine the precise form of $gMg^{-1}\cap N$, in particular to determine for a given $g$ whether or not $g \in NM$. These strong results that follow from Newman’s work underline why one-relator groups with torsion are easier to work with than one-relator groups in general.

has been significantly extended and generalised by 