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 CG(a)) is the set of elements of G which commute with a; in other words, CG(a) = {x in G : xa = ax}. If H is a subgroup of G, then CH(a) = CG(a) ∩ H. If there is no danger of ambiguity, we can write CG(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 −1s = xsy −1 = sxy −1.
The center of a group G is CG(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 NG(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 −1cn 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 −1t = sn −1. So
- (n −1cn)s = (n −1c)tn = (n −1(tc)n = (sn −1)cn = s(n −1cn)
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 NG(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) = Tx(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)).