nLab
centralizer

Contents

Definition

The centralizer subgroup of a subset SS in a group GG is the set C G(S)C_G(S) of all elements cGc\in G such that cs=scc s=s 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.

Revised on December 11, 2015 04:49:12 by Urs Schreiber (78.104.9.204)