# What does Sylow theory have to say about group presentations?

What does Sylow theory have to say about group presentations?

Of the books on 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.

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

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.

• Actually, Matt's comment is on point. You have a presentation for a group $G$ and you would like to know, for example: can we deduce anything about the Sylow $p$-subgroups of $G$? The first thing you'd like to know is the order of $G$, provided that $G$ is finite (since e.g. if $|G| = p^3qrs$ for some primes $p, q, r, s$ then you know the structure--and also a presentation for--the Sylow $t$-subgroups of $G$, where $t \in \{q,r,s\}$). But the problem of finding the order of $G$, given a presentation for it, is undecidable, so the question (in its generality) is most likely hopeless. – the_fox Apr 23 at 21:06