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 CG(a)) is the set of elements of G which commute with a; in other words, CG(a) = {x ∈ 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). 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 ∈ G : ∀ s ∈ 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 for all s in S.
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 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 ∈ 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 NG(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 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)).
