nLab
centralizer

Contents

Contents

Definition

The centralizer subgroup of a subset SS in (the set underlying) a group GG is the subgroup

C G(S)G C_G(S) \subset G

of all elements cGc \in G such that cs=scc \cdot s = s \cdot c for all sSs \in S.

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

The centralizer is the largest subgroup HH of GG containing SS such that SS is in the center of HH. The centralizer of a subset is clearly a subgroup of its normalizer, as fixing the set gH=Hgg H=H g is a weaker requirement than gh=hgg h=h g for all hHh\in H.

References

See also

Last revised on March 31, 2019 at 12:55:20. See the history of this page for a list of all contributions to it.