Finite Group Theory - 2014-12-19

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8:05 AM

If you want a daily problem, here is one (a nice and somewhat tricky Sylow theorem problem). Let $G$ be a finite group such that for any proper subgroup $H< G$ there is a prime $p$ such that $H$ is a $p$-group. Show that $G$ is not simple.

Alexander Gruber

8:31 AM

@TobiasKildetoft Lately I've been doing some cool stuff man

I've been looking at prime complexes of finite groups and making a whole lot of headway.

Tobias Kildetoft

@AlexanderGruber Cool

Alexander Gruber

I came up with a conjecture I'm trying to prove:

Tobias Kildetoft

what is a prime complex?

Alexander Gruber

@TobiasKildetoft Denote by $\pi(G)$ the set of prime divisors of $|G|$. The prime complex of a group is the set of subsets $P$ of $\pi(G)$ such that $G$ has a subgroup isomorphic to $C_{\prod_{p\in P}}$

Tobias Kildetoft

Alexander Gruber

8:43 AM

So, it's the same thing as the prime graph, except instead of edges (2-subsets) we've got subsets of any size

Tobias Kildetoft

sounds like a simplicial set of some sort

So are you using geometrical methods to study this, or purely group theoretic ones?

Alexander Gruber

@TobiasKildetoft that's precisely what it is

@TobiasKildetoft Right now I'm just using it as a language to talk about the questions I want to talk about

i.e. only group theory is proving things, i'm just using simplicial juju as words to talk about what I'm doing

Tobias Kildetoft

ahh, ok

Alexander Gruber

but I do expect that eventually I'll be able to use some results or methods from simplicial stuff to prove things about the group

for example, when I was doing prime graphs, it was just a language for a long time, but eventually I was able to use some basic Ramsey theory to get a quick proof on something that would have been monstrous otherwise.

Tobias Kildetoft

cool

Alexander Gruber

8:56 AM

so, speaking of

this is the conjecture I think is true:

**Definition.** Let $G$ be solvable with order divisible by exactly three distinct primes, $p$, $q$, and $r$. We call $G$ *sneaky* if $G$ contains subgroups isomorphic to $C_{pq}$, $C_{pr}$, $C_{qr}$, but no subgroup isomorphic to $C_{pqr}$. If $G$ has no sneaky proper subgroups, then $G$ is minimally sneaky.

**Conjecture.** Minimally sneaky groups are decomposible (i.e. they are direct products).

Tobias Kildetoft

interesting

Balarka Sen

cayley graph of $A_5$ looks like an icosahedron truncated along the vertices and $A_5$ is precisely the symmetry group of an icosahedron. is this merely a coincidence?

i think this has something to do with the fact that $A_5$ acts transitively on the vertices of the graph.

so yeah i guess groups come up as symmetry groups of the corresponding graphs.

Tobias Kildetoft

@AlexanderGruber So being sneaky means there is a triangle in the complex with no corresponding 2-simplex

Alexander Gruber

@TobiasKildetoft Yes, precisely.

Balarka Sen

what's a prime complex?

oh ok just saw that

Alexander Gruber

9:12 AM

Hence the name. You see a triangle in the prime graph, you want to think there'll be an element that is the product of all of the primes. If there isn't, that is sneaky.

Tobias Kildetoft

indeed

hmm, I would have guessed that the product of non-sneaky groups was again non-sneaky

Balarka Sen

in general probably extension of nonsneaky groups by nonsneaky groups are nonsneaky.

is it?

Alexander Gruber

example: $\left(C_{q_1}\rtimes C_p\right)\times \left(C_{q_2}\rtimes C_p\right)$

where the $\rtimes$ are Frobenius

Tobias Kildetoft

ahh

Balarka Sen

this is interesting

what happens at the graph level when we take products?

Alexander Gruber

9:16 AM

@BalarkaSen You get complete bipartite-ness

(assuming the set of primes is disjoint anyhow)

@TobiasKildetoft you can also make that type of construction with $2$-Frobenius groups, e.g. $S_4\times C_5\rtimes C_2$

But the thing is, those are the only examples I can come up with. So I've been thinking they are the only examples.

i.e. there aren't any indecomposible groups that can trick me with their sneaky prime graphs

Tobias Kildetoft

@AlexanderGruber Have you tried letting GAP run through a bunch of groups?

Alexander Gruber

@TobiasKildetoft Yup. holds for every group GAP can check

Tobias Kildetoft

how far up is it that is with the smallgroups library?

Alexander Gruber

(well, I mean, every preprogrammed group, all the SmallGroups plus the other easy libraries)

everything up to 2000 except 1024, and then cube-free groups up to some big number

Tobias Kildetoft

yeah, fortunate that you did not need to check those with just two prime divisors

Alexander Gruber

9:24 AM

One of the reasons prime graphs are a bit in vogue, I think, is that the orders get bigger than GAP can look at pretty quick for anything that's not basic.

Tobias Kildetoft

I had to give up checking a conjecture for order 768

Alexander Gruber

Hahahaha yeah 768 is a killer

Tobias Kildetoft

but then, for another I could go no higher than 384 before GAP could not assign more memory

though probably that code could be improved by applying some other results first, rather than checking naively each time

Alexander Gruber

@TobiasKildetoft When I was first looking at that one $G\times G\rightarrow H\times H$ problem I ran a GAP search that got locked up at something like order 12 :p

Tobias Kildetoft

heh

Balarka Sen

9:30 AM

hmm @AlexanderGruber. those groups have disconnected prime graphs.

speaking of it, is there a characterization of groups with disconnected prime graphs.

Alexander Gruber

@BalarkaSen There sure is

Balarka Sen

ah?

Alexander Gruber

solvable groups with disconnected prime graphs are either Frobenius groups or $2$-Frobenius groups.