# 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.

• My two cents: although I know little about combinatorial group theory, I would not expect any theorems of the sort you are looking for to exist. The fact that standard books on the subject make no mention of such theorems is, perhaps, evidence in favour of that claim. Still, something might be known, but I would expect any such results to be present in research-level papers. – the_fox Apr 23 '19 at 20:52
• So I'm not a group theorist, but in my humble opinion the reason you're finding no connection is because the presentation does not lend itself to finding Sylow subgroups. There is also no algorithm that, given a presentation, can always correctly determine whether the group is finite, as that is an undecidable problem. – Matt Samuel Apr 23 '19 at 20:54
• Echoing the other commenters, this seems like way, WAY too much to ask. – Randall Apr 23 '19 at 20:56
• On MO it may sit around for awhile but someone should be able to definitively tell you whether such a thing is known. – Matt Samuel Apr 23 '19 at 21:02
• 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 '19 at 21:06

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, $$0\le r, and $$m$$ and $$n(r-1)$$ are coprime.