Centralizer and normalizer

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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.

[edit] 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). Another, less common, notation is sometimes used when there is no danger of ambiguity, namely, Z(a), which parallels the notation for the center of a group. With this latter notation, one must be careful to avoid confusion between the center of a group G, Z(G), and the centralizer of an element g in G, given by Z(g).

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 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 its name from the fact that if S is a subgroup of G, then N(S) is the largest subgroup of G having S as a normal subgroup. The smallest normal subgroup of G containing <S> is called its conjugate closure.

A subgroup H of a group G is called a self-normalizing subgroup of G if N_G(H) = H.

[edit] 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)).