Finite groups are covered by their maximal centralizers and by their minimal centralizers; nonabelian p-groups that are F-groups have a number of distinct nontrivial centralizers congruent to 1 modulo p.
Sun, Finite covers of groups by cosets or subgroups,Internat
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Covering by Centralizers
Finite groups are covered by their maximal centralizers and by their minimal centralizers; nonabelian p-groups that are F-groups have a number of distinct nontrivial centralizers congruent to 1 modulo p.