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A note on maximal subgroups in finite groups. (English) Zbl 0745.20019
Here two results are proved. (1) If a finite group \(G\) has exactly two maximal subgroups, then \(G\) is cyclic and \(| G|\) is divisible by two distinct primes. (2) A group \(G\) which has exactly three maximal subgroups and is not a group of prime power order is necessarily cyclic and its order is divisible by at most three primes.
MSC:
20D15 Finite nilpotent groups, \(p\)-groups
20D25 Special subgroups (Frattini, Fitting, etc.)
20E28 Maximal subgroups
20D30 Series and lattices of subgroups
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