A Lower Bound on the Number of Maximal Subgroups in a Finite Group
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A Lower Bound on the Number of Maximal Subgroups in a Finite Group Jiakuan Lu1 · Shenyang Wang1 · Wei Meng2 Received: 24 October 2019 / Accepted: 20 December 2019 © Iranian Mathematical Society 2020
Abstract For a finite group G, let m(G) denote the set of maximal subgroups of G and π(G) denote the set of primes which divide |G|. In this paper, we prove a lower bound on |m(G)| when G is not nilpotent, that is, |m(G)| ≥ |π(G)| + p, where p ∈ π(G) is the smallest prime that divides |G| such that the Sylow p- subgroup of G is not normal in G. Keywords Finite group · Maximal group Mathematics Subject Classification 20D10 · 20D20
1 Introduction Throughout the following, all groups considered are finite. G always denotes a finite group. Let m(G) denote the set of maximal subgroups of G and π(G) denote the set of primes which divide |G|. The upper bounds of |m(G)| have previously been investigated. Wall [3] conjectured that |m(G)| < |G| and proved this conjecture when G is solvable. It is clear that |m(G)| = |π(G)| if and only if G is cyclic. Lauderdale
Communicated by Hamid Mousavi.
B
Jiakuan Lu [email protected] Shenyang Wang [email protected] Wei Meng [email protected]
1
School of Mathematics and Statistics, Guangxi Normal University, Guilin 541004, Guangxi, People’s Republic of China
2
School of Mathematics and Computer Science, Yunnan Minzu University, Kunming 650504, Yunnan, People’s Republic of China
123
Bulletin of the Iranian Mathematical Society
[2] proved |m(G)| ≥ |π(G)| + p, where p ∈ π(G) is the smallest prime such that the Sylow p-subgroup of G is noncyclic. In this paper, we prove another lower bound for |m(G)| and have the following result. Theorem 1.1 Let G be a nonnilpotent group. Assume that p ∈ π(G) is the smallest prime such that P ∈ Syl p (G) is not normal in G. Then, |m(G)| ≥ |π(G)| + p. Suppose that G is nilpotent and let G = P1 × P2 × · · · × Pn , where Pi are Sylow subgroups of G. Then |m(G)| = |m(P1 )| + |m(P2 )| + · · · + |m(Pn )|. When G is not nilpotent, |m(G)| ≥ |m(P1 )| + |m(P2 )| + · · · + |m(Pn )| + p need not be true, where p ∈ π(G) is the smallest prime such that P ∈ Syl p (G) is not normal in G. For example, let G = A4 , the alternating group of degree four, and let P2 and P3 be the Sylow 2 and 3-subgroups of G, respectively. Then |m(P2 )| + |m(P3 )| + p = 3 + 1 + 3 = 7; however, |m(G)| = 5. Corollary 1.2 Let G be a nonsolvable group and p be the smallest prime in π(G). Then |m(G)| ≥ |π(G)| + p. The notation is standard and follows that of Gorenstein [1].
2 Proofs We begin with some lemmas needed to prove the main theorem. Lemma 2.1 Let q be the largest prime in π(G) and assume that Q ∈ Sylq (G) is not normal in G. Then there are at least q + 1 maximal subgroups of G which contain N G (Q). Proof Since Q is not normal in G, we have that N G (Q) = G. Let M be a maximal subgroup of G containing N G (Q) and set n = |G : M|. Then M is self-normalizing in G. Let G act by right multiplication on the set of right cosets of M in G. Then G/MG is isomorphic to a subgroup
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