Number of conjugacy classes in extension group is bounded by product of number of conjugacy classes in normal subgroup and quotient group

From Groupprops
Jump to: navigation, search

Statement

Statement in terms of conjugacy classes

Suppose G is a finite group and H is a normal subgroup of G with quotient group G/H. Denote by c(G),c(H),c(G/H) respectively the number of conjugacy classes in G,H,G/H respectively. Then, we have the relation:

c(G) \le c(H)c(G/H)

Statement in terms of conjugacy classes

Suppose G is a finite group and H is a normal subgroup of G with quotient group G/H. Denote by CF(G),CF(H),CF(G/H) respectively the commuting fractions of G,H,G/H respectively. Then, we have the relation:

CF(G) \le CF(H)CF(G/H)