Centralizer and normalizer

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 ^-1s = 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 releated 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 ^-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 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)).