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A note on minimal and maximal subgroups in a finite group. (English) Zbl 0686.20017

The paper is divided into two sections. The first part deals with the description of the finite groups G having exactly two or three minimal subgroups. A group G which has two minimal subgroups turns out to be a \(p^ aq^ b\)-group with some technical conditions, while a group G with three minimal subgroups is either a 2-group with \(\Omega_ 1(G)\cong Z_ 2\times Z_ 2\) or a \(p^ aq^ br^ c\)-group with certain supplementary conditions.
The second part of the paper classifies the finite groups having two or three maximal subgroups. Those with two maximal subgroups are cyclic of order pq (p\(\neq q)\) and those with three maximal subgroups are either cyclic groups of order pqr (p\(\neq q\neq r\neq p)\) or the Klein four group.
Reviewer’s remark: Th structure of finite groups having at most three maximal subgroups was known to G. A. Miller, who determined these groups [in Proc. Natl. Acad. Sci. USA 27, 68-71 (1941)].
Reviewer: M.Deaconescu

MSC:

20D30 Series and lattices of subgroups
20E28 Maximal subgroups
20D20 Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure
20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks
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