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|. When G is a cyclic group, an elementary calculation proves that |m(G)| = |π(G)|. In this paper, we prove lower bounds on |m(G)| when G is not cyclic. In general, \({|m(G)| \geq |\pi(G)|+p}\) , where \({p \in \pi(G)}\) is the smallest prime that divides |G|. If G has a noncyclic Sylow subgroup and \({q \in \pi(G)}\) is the smallest prime such that \({Q \in {\rm syl}_q(G)}\) is noncyclic, then \({|m(G)| \geq |\pi(G)|+q}\) . Both lower bounds are best possible.
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Lauderdale, L.K. Lower bounds on the number of maximal subgroups in a finite group. Arch. Math. 101, 9–15 (2013). https://doi.org/10.1007/s00013-013-0531-2
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DOI: https://doi.org/10.1007/s00013-013-0531-2