conjugate subgroup

- group, ∞-group
- group object, group object in an (∞,1)-category
- abelian group, spectrum
- group action, ∞-action
- representation, ∞-representation
- progroup
- homogeneous space

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.

See also

- Wikipedia,
*Conjugacy of subgroups and general subsets*

Last revised on March 30, 2019 at 05:31:48. See the history of this page for a list of all contributions to it.