On the nonabelian composition factors of a finite group that is prime spectrum minimal

N. Maslova,   D. Revin

We consider finite groups only.

Let \(G\) be a finite group. The set \(\pi(G)\) of all prime divisors of the number \(|G|\) is called the prime spectrum of \(G\). A finite group \(G\) is prime spectrum minimal if \(\pi(H)\ne\pi(G)\) for every proper subgroup \(H\) of \(G\).

A subgroup \(H\) of a group \(G\) is called a Hall subgroup if \((|H|,|G:H|)=1\). The class of finite groups in which all maximal subgroups are Hall is a proper subclass of the class of finite prime spectrum minimal groups.

P. Shumyatsky has written down to the Kourovka Notebook [1, Problem 17.125] the following

Conjecture 1. In any finite group \(G\), there is a pair \(a,b\) of it's conjugate elements such that \(\pi(G)=\pi(\langle a,b\rangle)\).

It is easy to see that the Shumyatsky conjecture is equivalent to the following

Conjecture 2. Every prime spectrum minimal group is generated by a pair of conjugate elements.

In [2], we have proved that any finite group with Hall maximal subgroups is generated by a pair of conjugate elements. Thus we have obtained a partial solution to Problem 17.125 in the Kourovka Notebook. In our proof, we used the description of nonabelian composition factors of a finite group with Hall maximal subgroups [3]. Thus, the following problem is of interest.

Problem. What are the nonabelian composition factors of finite groups that are prime spectrum minimal?

The main result of this talk is the following

Theorem. Let \(S=A_n\), where \(n \ge 5\). Then the following conditions hold:

\(\ \ \ (i)\ \ \) \(S\) is not isomorphic to a composition factor of a prime spectrum minimal group if \(n\) is not a prime;
\(\ \ (ii)\ \ \) \(S\) is a prime spectrum minimal group if \(n\) is a prime.

Following [4], let us define \(c(G)\) to be the least integer \(n\) such that there exist proper subgroups \(H_1, \ldots, H_{n}\) with \(\pi(G)=\pi(H_1) \cup \ldots \cup \pi(H_{n})\).

In [4], it was proved that \(c(A_n) \le 2\) for every \(n\).

Using the main theorem we obtain

Corollary. Let \(S=A_n\), where \(n \ge 5\). Then the following conditions hold:

\(\ \ \ (i)\ \ \) \(c(S)=1\) if \(n\) is not a prime;
\(\ \ (ii)\ \ \) \(c(S)=2\) if \(n\) is a prime.

Acknowledgement. The research was supported by RFBR (projects 13-01-00469 and 13-01-00505), by the Joint Research Program of UB RAS with SB RAS (project 12-C-1-10018) and with NAS of Belarus (project 12-C-1-1009), by the grant of the President of Russian Federation for young scientists (project MK-3395.2012.1), and by a grant from IMM UB RAS for young scientists in 2013.

References

1. V. D. Mazurov (ed.), E. I. Khukhro (ed.), The Kourovka Notebook. Unsolved problems in group theory. 17th ed., Sobolev Inst. Mat., Novosibirsk (2010), 136 p.
2. N. V. Maslova, D. O. Revin, Generation of a finite group with Hall maximal subgroups by a pair of conjugate elements, Trudy Inst. Mat. Mekh. UrO RAN, 19, N3 (2013). (Russian, to appear)
3. N. V. Maslova, Nonabelian composition factors of a finite group whose all maximal subgroups are Hall, Sib. Math. J., 53, N5 (2012), 853–861.
4. V. A. Belonogov, On sets of subgroups of a finite simple group that control its prime spectrum, Trudy Inst. Mat. Mekh. UrO RAN, 19, N3 (2013). (Russian, to appear)

See also the authors' pdf version: