Finite Non-abelian Groups Whose Non-abelian Subgroups Have Minimum Centralizers

Abstract

A finite non-abelian group G is called an ${\cal{M}}{\cal{C}}$-group if all non-abelian subgroups H of G have minimum centralizers (i.e., C_G(H) = Z(G)). In this paper, the authors give some characterizations of ${\cal{M}}{\cal{C}}$-groups, and it is proved that ${\cal{M}}{\cal{C}}$-groups are just the finite groups with modular centralizer lattice of length 2 depicted by Schmidt, which leads to a classification of ${\cal{M}}{\cal{C}}$-groups. However, Schmidt’s depiction said nothing for ${\cal{M}}{\cal{C}}$-p-groups. They give a characterization of ${\cal{M}}{\cal{C}}$-p-groups. In particular, they characterize special ${\cal{M}}{\cal{C}}$-p-groups by means of the commutator matrices, and provide a method to determine or classify special ${\cal{M}}{\cal{C}}$-p-groups. As applications, some examples are given, and special ${\cal{M}}{\cal{C}}$-p-groups with an abelian maximal subgroup are classified up to isoclinism.

Keywords

Centralizers of groups / Finite p-groups / Special p-groups / Isoclinism / Commutator matrix / 20D15 / 20D30 / 05C25

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