Search Results

The papers in the accepted answer by @SixWingedSeraph all refer to a somewhat more specific problem, that of extending an automorphism of a normal subgroup to a larger group. Although the questioner …

Have you seen the paper The Frattini subgroup of a direct product of groups? Theorem 1 of that paper gives a necessary and sufficient condition for failure of the equality $\Phi(G \times H) = \Phi(G) …

Here's a short answer, assuming only that $C$ is an abelian $p$-group and $\vert D \vert$ is relatively prime to $p$. Remember that $G \cong H\rtimes K$ iff normal subgroup $H$ has a complement in $G …

You can handle several of the groups on the list from your edit with a modification of the argument of Marty Isaacs. For the rest, you can find guidance as to what a counterexample would look like. …

This has the content of a comment, but is somewhat longer and with more formatting than a comment allows, so I post it as an answer. Anyway: I tried the GAP computation for 2^4, and got results. Fo …

Although you say you'd prefer not to use GAP, producing a Hasse diagram is very easy in GAP, at least with the right packages. You'll need the xgap GAP package; and either the xgap binaries, which re …

A pretty complete and accessible description of this can be found in a survey article of Oliver King. In addition to the link, here's the citation info: King, Oliver H., The subgroup structure of fi …

You might find helpful the paper of Liebeck and Shalev "Classical Groups, Probabilistic Methods, and the (2,3)-Generation Problem". There, they show that there are three involutions that generate all …

Carlisle King recently posted an arXiv preprint which (claims to) show that every finite simple group is generated by an involution, together with another element of prime order. http://arxiv.org/abs/ …

As @Keith Kearnes says, the negative answer ought to be somewhere in Roland Schmidt's book. Unless I'm mistaken, it suffices to find two non isomorphic groups with isomorphic coset lattices. Indeed, …

UPDATE: The original poster of the question, together with Mamta Balodi, have shown that the labeling I suggest below is an EL-labeling if and only if group (product) complements coincide with lattic …

In a somewhat different direction from Alireza: the conjecture is true for a large family of groups, including all abelian groups and many supersolvable groups. Let me start with the abelian case. P …

Are you running computer experiments to verify conjectures? If so, the GAP SmallGroups library will give you exactly what you want up to $n = 1023$. For example, the GAP commands n:=16;; G:=SmallGr …