Number of conjugacy classes in a subgroup may be more than in the whole group

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Statement

It is possible to have a finite group G and a subgroup H of G such that the Number of conjugacy classes (?) in H is more than in G.

Related facts

Similar facts

Opposite facts

Proof

Example of the dihedral group of degree five

Further information: dihedral group:D10, cyclic group:Z5

The smallest pair of examples is where the group G is D_{10}, the dihedral group of degree five and order ten, and H is the subgroup is cyclic group:Z5. The group D_{10} has four conjugacy classes: one of involutions, one identity element, and two conjugacy classes in the cyclic subgroup of order five. On the other hand, H is an abelian group of order five hence has five conjugacy classes.

Other dihedral examples

Further information: element structure of dihedral groups

More generally, for odd n, the dihedral group of order 2n and degree n has (n + 3)/2 conjugacy classes, and the cyclic subgroup of order n has n conjugacy classes. For n \ge 5, the cyclic subgroup has more conjugacy classes than the whole group.

For even n, the dihedral group of order 2n has (n + 6)/2 conjugacy classes and the cyclic subgroup of order n has n conjugacy classes. For n \ge 8, the cyclic subgroup has more conjugacy classes than the whole group. The first example of this is cyclic group:Z8 in dihedral group:D16.