On control of the prime spectrum of a finite simple group

V. Belonogov

Let \(G\) be a finite group. We call the set \(\pi(G)\) of all prime divisors of \(G\) the \(\pi\)-spectrum or the prime spectrum of \(G\).

We say that the sections (in particular, subgroups) \(H_1,\dots, H_m\) of \(G\) control the prime spectrum of \(G\) (or control \(\pi(G)\)) if \[ \pi(H_1)\cup\dots\cup\pi(H_m)=\pi(G). \tag{1} \] We denote by \(c(G)\) the minimal of the numbers \(m\) for which the above situation is possible for some proper subgroups (or homomorphic images of proper subgroups) \(H_i\) of \(G\). It is clear that the definition of \(c(G)\) does not depend on whether we choose for \(H_i\) in (1) proper subgroups or arbitrary homomorphic images of proper subgroups of \(G\). The parameter \(c(G)\) is not defined only when \(G\) is the identity group or a group of prime order.

In the author's article [1] (in preparation), some properties of finite simple nonabelian groups connected with the above notions are studied. Situations of specific interest arise when the number \(m\) in (1) is small (for example, \(m=c(G)\)), or moreover when the subgroups \(H_1,\dots, H_m\) are chosen with some special "good" structure convenient for applications to a given particular investigation.

In [1], for every finite simple nonabelian group \(G\), some the set \(\{H_1,\dots, H_m\}\) of homomorphic images of proper subgroups of \(G\) satisfying condition (1) is indicated. In any case, \(m\le5\) and each of the sections \(H_1,\dots, H_m\) is a simple nonabelian group, Frobenius group, or (in a single case) a dihedral group. Precisely such selection of sections \(H_i\) turn out be useful in a certain concrete situation considered in the last section of [1], see also [2].

In this work, some corollaries of the results of [1] are given. Note that \(c(G/\Phi(G))=c(G)\), where \(\Phi(G)\) is the Frattini subgroups of \(G\), as \(\pi(G/\Phi(G))=\pi(G)\).

Henceforth, \(q\) denotes a prime power.

Theorem 1. If \(G\) is a finite alternating or classical simple group, then \(c(G)\le2\). In addition, \(c(G)=1\) if \(G\) is isomorphic to one of the groups: \(A_n\) for a nonprime \(n\), the groups \(PSp_{\,4}(q)\), \(P\Omega_{4n+1}(q)\) and \(P\Omega_{4n}^+(q)\) for all \(q\) and the groups \(PSL_6(2)\), \(PSU_3(3)\), \(PSU_3(5)\), \(PSU_4(2)\), \(PSU_4(3)\), \(PSU_5(2)\), \(PSU_6(2)\), \(PSp_6(2)\), \(\Omega_7(2)\).

Theorem 2. For any finite simple group \(G\) of exceptional Lie type, we have \(c(G)\le5\). Moreover,

\(c(G)=1\) if \(G\) is isomorphic to \(G_2(3)\) or group Tits \({}^2F_4(2)'\);
\(c(G)=2\) if \(G\) is isomorphic to \(G_2(q)\) for \(q>3\) or \({}^3D_4(q)\);
\(c(G)\le2\) if \(G\simeq F_4(q)\) \((\,c(G)=2\) for \((q,6)=1\,)\);
\(c(G)=3\) if \(G\) is isomorphic to \(Sz(q)\) for \(q>2\) or \({}^2G_2(q)\) for \(q>3\);
\(c(G)\le3\) if \(G\) is isomorphic to \(E_6(q)\), \({}^2E_6(q)\) or \(E_7(q)\);
\(c(G)=4\) if \(G\simeq {}^2F_4(q)\);
\(c(G)\le5\) if \(G\simeq E_8(q)\).

For sporadic simple groups \(G\), the question of the size of \(c(G)\) is completely settled in the following theorem.

Theorem 3. Let \(G\) be a finite sporadic simple group. Then \(c(G)\le5\). Namely,

\(c(G)=1 \Longleftrightarrow G\) is isomorphic to \(M_{11}\), \(M_{12}\), \(M_{24}\), \(HS\), \(Mc\), \(Co_3\) or \(Co_2\);
\(c(G)=2 \Longleftrightarrow G\) is isomorphic to \(M_{22}\), \(M_{23}\), \(J_{2}\), \(J_{3}\), \(He\), \(Suz\), \(Ru\), \(O'N\), \(Co_1\), \(Fi_{22}\), \(Fi_{24}'\) or \(F_5\);
\(c(G)=3 \Longleftrightarrow G\) is isomorphic to \(J_{1}\), \(Ly\), \(Fi_{23}\), \(F_3\) or \(F_2\);
\(c(G)=4 \Longleftrightarrow G\simeq F_1\);
\(c(G)=5 \Longleftrightarrow G\simeq J_4\).

Generally, if \(G\) is a nonsimple finite group then \(c(G)\le 2\).

Proposition 1. Let \(G\) be a finite group and \(c(G)\ge 3\). Then \(G/\Phi(G)\) is a simple nonabelian group and \(c(G)=c(G/\Phi(G))\).

In fact, if \(G\vartriangleright N>\Phi(G)\) then \(G=MN\) for a maximal subgroup \(M\) of \(G\) and, therefore, \(c(G)\le 2\).

Thus, from Theorems 1–3, the classification of the finite simple groups [3] and Proposition 1 it follows

Proposition 2. We have \[\{m \mid m=c(G) \ \ \text{for a finite group} \ \ G\} =\{1,2,3,4,5\}.\] As seen from Propositions 1 and 2, in the considered themes the finite simple nonabelian groups participate essentially.

For a finite solvable group \(G\), the upper bound \(c(G)=2\) is achieved if and only if every maximal subgroup of \(G\) is a Hall subgroup of \(G\). Finite groups \(G\) whose all maximal subgroups are Hall subgroups of \(G\) are investigated in [4,5].

In connection with the above results, the following problems may be of interest.

Problem 1. Find \(c(G)\) for every finite simple nonabelian group \(G\).

According to Theorem 1, for classical finite simple groups \(G\), to solve Problem 1 is the same as to describe such \(G\) with \(c(G)=1\).

Problem 2. For a given finite simple nonabelian group \(G\) (of a specific type), find all finite simple nonabelian groups \(H\) (also of a specific type) such that \(\pi(G)=\pi(H)\).

Problem 3. For a given set \(\mu\) of primes, find all finite simple nonabelian groups \(H\) (of a specific type) such that \(\pi(G)=\mu\).

For example, the following problem is a special case of Problems 2 and 3.

Problem 4. Find all finite simple nonabelian groups \(G\) such that \(\pi(G)=\pi(A_p)=\pi(p\,!)\) for a given prime \(p\).

Problem 5. For finite simple groups \(G\) with \(c(G)=2\), describe all pairs \(\{M,K\}\) of maximal subgroups of \(G\) such that \(\pi(G)=\pi(M)\cup\pi(K)\).

A similar problem may be also formulated for some groups with \(c(G)>2\).

Acknowledgement. The work is supported by RFBR (project 13-01-00469), RFBR–NSFC of China (project 12-01-91155), the Program of the Division of Mathematical Sciences of RAS (project 12-T-1-1003), and 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).

References

1. 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, in preparation)
2. V. A. Belonogov, Finite groups all of whose maximal subgroups are \(\pi\)-decomposable, 11th Internat. Conf. "Group Theory and its Applications", Abst., Vladikavkaz (2012), 126–127.
3. D. Gorenstein, R. Lyons, R. Solomon, The classification of the finite simple groups,Math. Surveys and Monographs., 40, N1 (1994), 165 p.
4. V. S. Monakhov, Finite \(\pi\)-solvable groups whose maximal subgroups have the Hall Property, Math. Notes, 84, N3 (2008), 390–394.
5. N. V. Maslova, Nonabelian composition factors of finite groups whose all maximal subgroups are Hall, Sib. Math. J., 53, N5 (2012), 853–861.

See also the author's pdf version: