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A note on solitary subgroups of finite groups. (English) Zbl 1351.20007
A subgroup \(H\) of a finite group \(G\) is called solitary (respectively, normal solitary) when it is a subgroup (respectively, normal subgroup) of \(G\) such that no other subgroup (respectively, normal subgroup) of \(G\) is isomorphic to \(H\). A normal subgroup \(N\) of a group \(G\) is said to be quotient solitary if no other normal subgroup \(K\) of \(G\) has a quotient isomorphic to \(G/N\). In this paper, some new results about lattice properties of these subgroups and their relation with classes of groups are given. The authors also present several interesting examples showing a negative answer to some questions about these subgroups.
MSC:
20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks
20D30 Series and lattices of subgroups
20F16 Solvable groups, supersolvable groups
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