Let \(G\) be a group, and let \(s\in G\). The centralizer of \(s\) in \(G\) is defined to be \[ C(s)=\{g\in G:gs=sg\}. \] (a) Prove that \(C(s)\) is a subgroup of \(G\). (b) Prove that \(\langle s\rangle\) is contained in \(C(s)\). (Recall that \(\langle s\rangle\) is the subgroup of \(G\) generated by \(s\).) (c) Prove that \(\langle s\rangle\) is a normal subgroup of \(C(s)\).
Solution
Proof. (a): (Nonempty): We have es =s =se, and therefore e∈C(s). So C(s)⁄= ∅. (Closed under∗): Given g,h∈C(s), we have (gh)s =g(hs) =g(sh) = (gs)h = (sg)h =s(gh), so gh∈C(s), as desired. (Closed under inverses): Given g∈C(s), we have gs =sg, so g−1s =g−1sgg−1 =g−1gsg−1 =sg−1, so g−1∈C(s), as desired. QED (a) (b): Given x∈⟨s⟩, we have x =sn for some n∈ Z. So xs =sns =sn+1 =ssn =sx, so x∈C(s), as desired. QED (b) (c): Given g∈C(s) and h∈⟨s⟩, there is some n∈ Z such that h =sn. So (ghg−1)s =gsng−1s =sngg−1s =sns =sn+1 =ssn =ssngg−1 =sgsng−1 =s(ghg−1). Thus,ghg−1∈C(s), as desired. QED
Other problems involve using the fact that a subgroup is normal to prove something else.