In group theory, the **centralizer** and **normalizer** of a subset *S* of a group *G* are subgroups of *G* which have a restricted action on the elements of *S* and *S* as a whole, respectively. These subgroups provide insight into the structure of *G*.
## Definitions
The **centralizer** of an element *a* of a group *G* (written as *C*_{G}(*a*)) is the set of elements of *G* which commute with *a*; in other words, C_{G}(*a*) = {*x* in *G* : *xa* = *ax*}. If *H* is a subgroup of *G*, then C_{H}(*a*) = C_{G}(*a*) ∩ *H*. If there is no danger of ambiguity, we can write C_{G}(*a*) as C(*a*). More generally, let *S* be any subset of *G* (not necessarily a subgroup). Then the centralizer of *S* in *G* is defined as C(*S*) = (*x* in *G* : for all *s* in *S*, *xs* = *sx*}. If *S* = {*a*}, then C(*S*) = C(*a*). C(*S*) is a subgroup of *G*; since if *x*, *y* are in C(*S*), then *xy*^{ -1}*s* = *xsy*^{ -1} = *sxy*^{ -1}. The *center* of a group *G* is C_{G}(*G*), usually written as Z(*G*). The center of a group is both normal and abelian and has many other important properties as well. We can think of the centralizer of *a* as the largest (in the sense of inclusion) subgroup *H* of *G* having having *a* in its center, Z(*H*). A related concept is that of the **normalizer** of *S* in *G*, written as N_{G}(*S*) or just N(*S*). The normalizer is defined as N(*S*) = {*x* in *G* : *xS* = *Sx*}. Again, N(*S*) can easily be seen to be a subgroup of *G*. The normalizer gets it name from the fact that if we let <*S*> be the subgroup generated by *S*, then N(*S*) is the largest subgroup of *G* having <*S*> as a normal subgroup (compare this with the conjugate closure of *S*).
## Properties If *G* is an abelian group, then the centralizer or normalizer of any subset of *G* is all of *G*; in particular, a group is abelian if and only if Z(*G*) = *G*. If *a* and *b* are any elements of *G*, then *a* is in C(*b*) if and only if *b* is in C(*a*), which happens if and only if *a* and *b* commute. If *S* = {*a*} then N(*S*) = C(*S*) = C(*a*). C(*S*) is always a normal subgroup of N(*S*): If *c* is in C(*S*) and *n* is in N(*S*), we have to show that *n*^{ -1}*cn* is in C(*S*). To that end, pick *s* in *S* and let *t* = *nsn*^{ -1}. Then *t* is in *S*, so therefore *ct* = *tc*. Then note that *ns* = *tn*; and *n*^{ -1}*t* = *sn*^{ -1}. So - (
*n*^{ -1}*cn*)*s* = (*n*^{ -1}*c*)*tn* = (*n*^{ -1}(*tc*)*n* = (*sn*^{ -1})*cn* = *s*(*n*^{ -1}*cn*) which is what we needed. If *H* is a subgroup of *G*, then the *N/C Theorem* states that the factor group N(*H*)/C(*H*) is isomorphic to a subgroup of Aut(*H*), the automorphism group of *H*. Since N_{G}(*G*) = *G*, the N/C Theorem also implies that *G*/Z(*G*) is isomorphic to Inn(*G*), the subgroup of Aut(*G*) consisting of all inner automorphisms of *G*. If we define a group homomorphism *T* : *G* → Inn(*G*) by *T*(*x*)(*g*) = *T*_{x}(*g*) = *xgx*^{ -1}, then we can describe N(*S*) and C(*S*) in terms of the group action of Inn(*G*) on *G*: the stabilizer of *S* in Inn(*G*) is *T*(N(*S*)), and the subgroup of Inn(*G*) fixing *S* is *T*(C(*S*)). |