What does Sylow theory have to say about group presentations?

Question

What does Sylow theory have to say about group presentations?

Of the books on combinatorial-group-theory I have looked in so far, the following do not contain any reference to Sylow's Theorems:

• Baumslag's "Topics in Combinatorial Group Theory",

• Collins et al.'s "Combinatorial Group Theory and Applications to Geometry",

• Lyndon et al.'s "Combinatorial Group Theory", and

• Stillwell's "Classical Topology and Combinatorial Group Theory"

to name but four (according to their indexes); and there's more; however,

• Coxeter et al.'s "Generators and Relations for Discrete Groups" says something about the presentations of things called $ZS$-metacyclic groups, which are groups whose commutator subgroup and commutator quotient group are cyclic, and all of whose Sylow subgroups are cyclic; and

• Sims' "Computation with Finitely Presented Groups" mentions Sylow's Theorems in historical notes, saying they date from 1872 and that they were, first, about finite permutation groups.

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I haven't found any link between presentations and Sylow's Theorems anywhere convenient online, although perhaps I haven't looked hard enough.

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What sort of thing am I looking for?

Well, perhaps, given a presentation $P$ of a group, say, some information about the presentations of its Sylow subgroups might be garnered from $P$. I don't know . . . Something like that anyway.

Please help :)

Answer 1

This is a partial answer.

In Robinson's, "A Course in the Theory of Groups (Second Edition)", Theorem 10.1.10 states the following.

Theorem (Hölder, Burnside, Zassenhaus) If $G$ is a finite group all of whose Sylow subgroups are cyclic, then $G$ has a presentation

$$G=\langle a,b\mid a^m=1=b^n, b^{-1}ab=a^r\rangle,$$

where $r^n\equiv 1\pmod{m}, m$ is odd, $0\le r<m$, and $m$ and $n(r-1)$ are coprime.

Conversely in a group with such a presentation all Sylow subgroups are cyclic.