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Number of conjugacy classes of subgroups

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Definition

The number of conjugacy classes of subgroups of a group is defined as follows:

  1. It is the number of conjugacy classes of subgroups, i.e., the number of equivalence classes in the collection of subgroups of the group under the equivalence relation of being conjugate subgroups in the whole group.
  2. It is the free rank of the Burnside ring of the group as a free abelian group.

For a finite group, this number is finite. For infinite groups, it is usually infinite, with some exceptions such as Tarski monsters.