On the Number of Conjugacy Classes of Subgroups of a Finite Group

Source abstract

Abstract Let $$k'(G)$$ k ′ ( G ) and L ( G ) be the number of conjugacy classes of subgroups and the subgroup lattice of a finite group G , respectively. Our objective is to study some aspects related to the ratios $$d'(G)=\frac{k'(G)}{|L(G)|}$$ d ′ ( G ) = k ′ ( G ) | L ( G ) | and $$d^*(G)=\min \{ d'(S) \mid S\text { is a section of }G\}$$ d ∗ ( G ) = min { d ′ ( S ) ∣ S is a section of G } which measure how close G is to being a Dedekind group. We prove that the set containing the values $$d'(G)$$ d ′ ( G ) , as G ranges over the class of nilpotent groups, is dense in [0, 1]. A nilpotency criterion is obtained by proving that if $$d^*(G)>\frac{2}{3}$$ d ∗ ( G ) > 2 3 , then G is nilpotent and information on its structure is given. We also show that if $$d^*(G)>\frac{4}{5}$$ d ∗ ( G ) > 4 5 , then G is an Iwasawa group. Finally, we deduce a result which ensures that a p -group of order $$p^n$$ p n ( $$n\ge 3$$ n ≥ 3 ) is a Dedekind group. This last result can be extended to the class of nilpotent groups and it also highlights the second maximum value of $$d^*$$ d ∗ on the class of p -groups of order $$p^n$$ p n .