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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
##### Keywords:
$$p$$-groups; finite group; maximal subgroups; cyclic