centralizer

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

The **centralizer subgroup** (also: *commutant*) of a subset $S$ in (the set underlying) a group $G$ is the subgroup

$C_G(S) \subset G$

of all elements $c \in G$ which commute with $S$, hence such that $c \cdot s = s \cdot c$ for all $s \in S$.

Notice the similarity with but difference to the concept of *normalizer subgroup*.

The centralizer of $S$ is clearly a subgroup of its normalizer

$C_G(S)
\subset
N_G(S)$

as fixing the set $g H = H g$ is a weaker requirement than $g h=h g$ for all $h\in H$.

See also

- Wikipedia,
*Centralizer and normalizer*

Last revised on November 6, 2020 at 15:00:13. See the history of this page for a list of all contributions to it.