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Encyclopedia > 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 CG(a)) is the set of elements of G which commute with a; in other words, CG(a) = {x in 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).


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 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 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 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 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)).


  Results from FactBites:
 
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The normal distribution also arises in many areas of statistics: for example, the sampling distribution of the mean is approximately normal, even if the distribution of the population the sample is taken from is not normal.
In that case, the assumption of normality is not justified, and it is the logarithm of the variable of interest that is normally distributed.
Normality is the central assumption of the mathematical theory of errors.
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In computing, a normal number is one that is within the normal range of a floating-point format.
In abstract algebra (in particular, group theory): a normal subgroup is a subgroup that is invariant under conjugation.
In functional analysis: a normal operator is a linear operator on a Hilbert space that commutes with its adjoint.
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