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.
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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;^{[1]} in other words, C_{G}(a) = {x ∈ 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 ∈ 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^{ −1}s = xsy^{ −1} = sxy^{ −1} for all s in S.
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 ∈ 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 normalizer should not be confused with the normal closure.
A subgroup H of a group G is called a selfnormalizing subgroup of G if N_{G}(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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