Centralizer and normalizer
Encyclopedia
In group theory
, the centralizer and normalizer of a subset
S of a group
G are subgroup
s 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.
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)s=x(ys) = (xs)y = s(xy) 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 : x-1Sx = S}. 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.
, 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. Sos = (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 and CG(G) = Z(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 automorphism
s 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)).
Group theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
, the centralizer and normalizer of a subset
Subset
In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...
S of a group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
G are subgroup
Subgroup
In group theory, given a group G under a binary operation *, a subset H of G is called a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H x H is a group operation on H...
s 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.
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 subgroupSubgroup
In group theory, given a group G under a binary operation *, a subset H of G is called a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H x H is a group operation on H...
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)s=x(ys) = (xs)y = s(xy) 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
Normal subgroup
In abstract algebra, a normal subgroup is a subgroup which is invariant under conjugation by members of the group. Normal subgroups can be used to construct quotient groups from a given group....
and abelian
Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
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 : x-1Sx = S}. 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
Normal subgroup
In abstract algebra, a normal subgroup is a subgroup which is invariant under conjugation by members of the group. Normal subgroups can be used to construct quotient groups from a given group....
. The normalizer should not be confused with the normal closure
Conjugate closure
In group theory, the conjugate closure of a subset S of a group G is the subgroup of G generated by SG, i.e. the closure of SG under the group operation, where SG is the conjugates of the elements of S:The conjugate closure of S is denoted or G.The conjugate closure of any subset S of a group G...
.
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 groupAbelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
, 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
Normal subgroup
In abstract algebra, a normal subgroup is a subgroup which is invariant under conjugation by members of the group. Normal subgroups can be used to construct quotient groups from a given group....
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. Sos = (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
Group isomorphism
In abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations. If there exists an isomorphism between two groups, then the groups are called isomorphic...
to a subgroup of Aut(H), the automorphism group of H.
Since NG(G) = G and CG(G) = Z(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 automorphism
Inner automorphism
In abstract algebra an inner automorphism is a functionwhich, informally, involves a certain operation being applied, then another one performed, and then the initial operation being reversed...
s of G.
If we define a group homomorphism
Group homomorphism
In mathematics, given two groups and , a group homomorphism from to is a function h : G → H such that for all u and v in G it holds that h = h \cdot h...
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
Group action
In algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...
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)).