Number of conjugacy classes in a subgroup of finite index is bounded by index times number of conjugacy classes in the whole group

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Statement

Statement in terms of number of conjugacy classes

Suppose G is a group and H is a subgroup of finite index. Suppose the number of conjugacy classes in G is a finite number c and the index of H in G is d. Then, the number of conjugacy classes in H is at most cd.

Statement in terms of commuting fraction

Suppose G is a finite group and H is a subgroup of finite index. Suppose the commuting fraction of G is a finite number \mu and the index of H in G is d. Then, the commuting fraction of H is at most d^2\mu.

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