nLab conjugate subgroup

Definition

For $G$ a group and $H_1, H_2 \hookrightarrow$ two subgroups, one says that they are conjugate subgroups if there exists an element $g \in G$ such that the conjugation action by $g$ takes $H_1 \to H_1$:

$$H_1 \sim_{conj} H_2 \;\;\; \coloneqq \;\;\; H_2 = Ad_g(H_1) = g H_1 g^{-1} \,.$$

The equivalence classes under this relation are hence called the conjugacy classes of subgroups of $G$. These play a key role in much of group theory and representation theory, for instance as parameters for group characters.

Related concepts

• conjugacy class

• subgroup lattice

• HNN-extension

References

Textbook account:

• Mark A. Armstrong, chapter 14 of: Groups and Symmetry, Undergraduate Texts in Mathematics, Springer (1988) [doi:10.1007/978-1-4757-4034-9, pdf]

See also

• Wikipedia, Conjugacy of subgroups and general subsets