nLab
centralizer

Definition

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

$$C_G(S) \subset G$$

of all elements $c \in G$ such that $c \cdot s = s \cdot c$ for all $s \in S$.

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

The centralizer is the largest subgroup $H$ of $G$ containing $S$ such that $S$ is in the center of $H$. The centralizer of a subset is clearly a subgroup of its normalizer, as fixing the set $g H=H g$ is a weaker requirement than $g h=h g$ for all $h\in H$.

Related concepts

• stabilizer subgroup

• normalizer subgroup

References

See also

• Centralizer and normalizer