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 can provide insight into the structure of G.

Contents

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;[1] in other words, CG(a) = {xG : 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) = {xG : ∀ sS, 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) = {xG : 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 normalizer should not be confused with the normal closure.

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

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).

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