This post comes directly from a talk of Elisa Covato at the seminar of the the PhD students of Bristol.

Definition of the prime graph of a group
Let G be a finite group. Let \pi (G) be the set of prime factors of \lvert G\rvert. The prime graph \Gamma(G) has \Pi(G) as vertex set and two primes r,s are neighbours if there exists g\in G whose order is multiple of rs.

Note that Lagrange’s theorem implies that every order of an element of G is a divisor of \lvert G\rvert, hence is a product of elements of \Pi(G) ; and that thanks to Cauchy, we know that every element of \Pi(G) is the order of somebody. But a product rs of elements of this set is not always the order of somebody, not even a divisor of the order of somebody.

In the case of the unique abelian group of order p_1\cdots p_k where the p_i are pairwise distinct primes, the prime graph is the complete graph.

In the case of \mathcal{A}_{10}, we have \Pi(G)=\{2,3,5,7\}. And there’s somebody of order 6, for instance (123)(45)(67) ; of order 10, for instance (12345)(67)(89). There’s also somebody of order 15, for example (12345)(678). There’s somebody of order 21, (1234567)(89A) (où A=10). But no one has order 14 nor 35. Indeed, recall that the order of a permutation is the ppcm of the orders of the cycles in its decomposition in a product of disjoint cycles ; and if 14 is the ppcm of some numbers, then 2 and 7 ; and in order to make an even permutation with at least one cycle of length 2 and at least one of length 7, you need 11 elements. For 5\times 7, you need 12 elements.

A lot of information about G is encoded in the prime graph but not every thing.

Recognition problem
Can a group G be uniquely determined by its prime graph (at least among a more or less restricted class of groups) ? So if \Gamma (G_1)=\Gamma(G_2) are the groups isomorphic ? Be careful, when I say that the graphs are equal, I mean it ! Not only isomorphic ! Otherwise for instance all the \mathbb{Z}_P have the same graph, a single vertex.

The answer is rather no in general.

Theorem (Lucchini, Morigi, Shumyatski)
Let G be a finit group. Then there exists H\leq G which is 3-generated and has the same prime graph.

Of course, the theorem doesn’t claim that H\neq G so it’s only interesting when G is not himself 3-generated (generated by a set of at most 3 elements). Moreover, if G is simple, it is even 2-generated (if you believe in the classification of finite simple groups).

Problem : let G be finite and simple. Determine (H,H) with H\leq G,\Gamma(H)=\Gamma(G).

Theorem (Burness, Covato)
Let G be finite, simple and H be a maximal subgroup. Then \Gamma (H)=\Gamma (G) implies that one of the following holds:
a) (G,H) belongs to a quite short table of exceptions.
b) G=\mathcal{A}_n,H=(\mathcal{S}_k\times\mathcal{S}_{n-k})\cap\mathcal{A}_n
Moreover, \Gamma(H)=\Gamma (G) whenever they are in the table.

I didn’t really get how the proof goes but apparently it uses a former result of Liebeck, Prager & Saxl who determined in which case the set of prime factors is the same. And then it uses an old result of Zsygmondi (1892) : for all prime power q, e integer, q^e has a prime divisor which divides no other q^i-1 with i\leq e.

When b) holds, it is not always true that the prime graphs are equal.

Conjecture : Suppose G=\mathcal{A}_n,H=(\mathcal{S}_k\times\mathcal{S}_{n-k})\cap\mathcal{A}_n with 1< k< n and p\leq n\implies p\leq k for every prime p.
Then \Gamma(H)=\Gamma (G)\Longleftrightarrow either
i)n\geq 25 is odd,k=n-1, n-4 is composite
ii)(n,k) is (6,5) or (10,7)

Goldbach's conjecture implies this conjecture. In fact
Lemma : suppose G=\mathcal{A}_{p+1}, H=\mathcal{A}_{p},p\geq 7. Then
\Gamma(H)=\Gamma (G)\Longleftrightarrow p+1=r+s.

We have restricted ourselves in all this to maximal subgroups H. Can we go further?
Theorem : let G be finite simple and H a proper subgroup. Let’s even suppose that G=\mathcal{A}_n and H is transitive (so we forbid the situation of b)). Then they have same prime graphs if and only if either
i) H is maximal and (G,H) are in the table.
ii) H is a second maximal group (maximal subgroup of maximal subgroup M).

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