nLab
centralizer

Contents

Contents

Definition

The centralizer subgroup (also: commutant) 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 which commute with SS, hence such that cs=scc \cdot s = s \cdot c for all sSs \in S.

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

The centralizer of SS is clearly a subgroup of its normalizer

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

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