nLab conjugate subgroup




For GG a group and H 1,H 2H_1, H_2 \hookrightarrow two subgroups, one says that they are conjugate subgroups if there exists an element gGg \in G such that the conjugation action by gg takes H 1H 1H_1 \to H_1:

H 1 conjH 2H 2=Ad g(H 1)=gH 1g 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 GG. These play a key role in much of group theory and representation theory, for instance as parameters for group characters.


See also

Last revised on December 11, 2022 at 09:32:36. See the history of this page for a list of all contributions to it.