On double centralizer subgroups of some finite 𝑝-groups

Abstract

Let A A be a maximal abelian normal subgroup of a finite p p -group G ( p > 2 ) G(p > 2) such that [ G , A ] [G,A] is cyclic. Then (i) C G ( C G ( D ) ) = D {C_G}({C_G}(D)) = D and [ G : C G ( D ) ] = [ D : Z ( G ) ] [G:{C_G}(D)] = [D:Z(G)] for every Z ( G ) ⩽ D ⩽ G Z(G) \leqslant D \leqslant G ; (ii) [ G : Z ( G ) ] = [ G , A ] 2 [G:Z(G)] = {[G,A]^2} and every faithful absolutely irreducible representation of G G is of degree [ G : A ] [G:A] . The case p = 2 p = 2 will also be mentioned.