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CONJUGATES, CONJUGATE CLASSES, AUTOMORPHISMS, NORMAL SUBGROUPS, QUOTIENT GROUPS, HOMOMORPHIC MAPPING OF GROUPS

Def. Transform. transform of an element of a group. Given group G. The transform Tx(a) of an element a ∈ G by an element x ∈ G is the element a' = x -1ax. The transform is a function or mapping that maps an element a ∈ G into an element a' = x -1ax ∈ G.

Conjugate elements in a group. Given group G. The transform a' = x -1ax of an element a ∈ G by an element x ∈ G is called the conjugate of a.

Condition for two elements of a group to be conjugate. Given group G. The element a ∈ G is conjugate to an element b ∈ G if there exists an element x ∈ G such that a = x -1bx. Let G be a group containing n elements. The set of all transforms of an element a ∈ G by all elements xi (i = 1, n) of G is the set of all conjugates of a in G and is a conjugate set (or class) in G.

We state this in a different way: Consider group G = {a1, a2, ... , an}. The element ai ∈ G is conjugate to an element aj ∈ G if there exists an element x ∈ G such that ai = x -1 aj x. The set of all transforms of an element ai G by all elements ai (i = 1, n) of G is the set of all conjugates of ai in G and is a conjugate set (or class) in G.  See Fig. 1. The set of elements in the highlighted line represent the set of conjugates of element ai. Each line of the table represents a different conjugate class in group G.

Conjugate classes. The relationship of conjugacy is a an equivalence relation defined on the elements of G:

1) Every element of a group is conjugate to itself (since a = a-1aa).

2) If a is conjugate to b then b is conjugate to a (because if b = x-1ax then xbx-1 = a or

a = (x-1)-1bx-1).

3) If a is conjugate to b and b is conjugate to c then a is conjugate to c ( because if b = x-1ax and c = y-1by , then c = y-1x-1axy = (xy)-1a(xy) ).

Thus conjugacy is a relation that is reflective, symmetric and transitive, making it an equivalence relation that splits a group up into disjoint equivalence classes of conjugate elements.

One can determine the elements in the separate conjugate classes as follows: Regard b = x-1ax as a transform that maps a into b. One can find all elements in G that are conjugate to a particular element a of G by determining the images of a for all elements xi ∈ G. By this process a conjugate class can be computed for each element aj ∈ G. It can be shown that each class obtained by this process will be either identical to others obtained or disjoint from them. The process will give all classes.

Def. Automorphism. An automorphism of a group G is a one-to-one mapping f :G → G (i.e. a one-to-one correspondence between elements of G) where f(ab) = f(a)f(b) for all a,b in G. It represents an isomorphism of group G with itself.

Theorem. For any fixed element x of a group G, the mapping Tx(a) : a → x -1ax carrying a into x -1ax effects an automorphism on G. Each x G gives a different automorphism.

Thus this theorem says that the mapping Tx(a) : a → x -1ax establishes a correspondence between members of G in which, for any selected x, f(ab) = f(a)f(b) for all a,b in G.

Proof. We need to prove that for any x in G, f(ab) = f(a)f(b) for all a,b in G. Thus we need to prove x-1(ab)x = (x-1ax)(x-1bx) for all a,b in G. The proof follows immediately since (x-1ax)(x-1bx) = x-1a(xx-1)bx = x-1abx.

Let H = {h1, h2, .... ,hn} be a subgroup of G. For a fixed x, apply the transform Tx(a) : a → x -1ax to all elements h1, h2, .... ,hn of H mapping them into elements k1, k2, .... ,kn. in G. The elements k1, k2, .... ,kn then correspond to a subgroup K of G that is isomorphic to H. Each element ki K is a conjugate of its counterpart hi H. Group K is said to be conjugate to H. Thus, in general, the transform Tx(a) : a → x -1ax effects a automorphic mapping from one subgroup of G into another subgroup of G. Each xi ∈ G gives a different automorphic mapping of group H, mapping H into another (or perhaps the same) subgroup of G. The set of all subgroups into which the transform Tx(a) : a →x -1ax maps H for all the different xi ∈ G is a set of subgroups conjugate to H. Any two of the subgroups are conjugate to each other.

Conjugate set of subgroups. Given group G and subgroup H. The set of different subgroups obtained by transforming a given subgroup H by all the elements of the group G is a conjugate set of subgroups; any two of these subgroups are conjugate to each other.

Inner and outer automorphisms.. Automorphisms Tx(a) : a → x -1ax on a group generated by the elements x are called inner automorphisms. All other automorphisms on a group are called outer automorphisms.

Note. If group G has n elements there will be n inner automorphisms Tx(a) : a → x -1ax , one for each element xi ∈ G (although some may be identical).

Self-conjugacy of elements. If the inner automorphism Tx(a) : a → x -1ax maps a into itself for all x ∈G , i.e. if a is sent into itself for every inner automorphism of G, a is called self-conjugate. The element a ∈ G is self-conjugate if and only if x -1ax = a or ax = xa for all x ∈ G, that is, if and only if a commutes with every element of G..

Conjugacy of complexes. The complex H of a group G is conjugate to the complex K of G if there exists an element x ∈G such that H = x -1Kx.

Thus, given a subgroup K, any subgroup H given by H = x -1Kx , for any x ∈ G, is conjugate to K.

Theorem. All conjugates of a subgroup G are subgroups of G.

Self-conjugacy of complexes. The complex H is self-conjugate if x -1Hx = H for all x ∈ G, that is, if the subset H is mapped into itself by every inner automorphism of G.

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Theorems.

1] An inner automorphism Tx(a) : a → x -1ax transforms a subgroup H into a subgroup x -1Hx, which is said to be a conjugate of H or conjugate to H.

2] If a subgroup H is identical with all its conjugate subgroups, i.e.

x -1Hx = H for every x,

it simply means that the subgroup H commutes with every element x i.e.

xH = Hx for every x,

and is, therefore, a normal divisor (or normal subgroup).

In other words, the subgroups invariant under all inner automorphisms are the normal divisors.

3] A subgroup is a normal divisor (or normal subgroup) if it contains with any element a all its conjugate elements x -1ax as well.

4] A subgroup H is a normal subgroup of G if and only if H is self-conjugate i.e. if x -1Hx = H for all x ∈ G (i.e. if H contains the conjugates of all its elements).

5] If H is a normal subgroup of a group G, then HK = KH for every complex K of G.

6] If H is a normal subgroup of G and K is any subgroup of G, then HK = KH is a subgroup of G.

7] A subgroup H is normal if and only if all its right cosets are equal to its left cosets.

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Quotient groups. The cosets of a normal subgroup H of a group G form a group under the operation of complex multiplication. This group formed by the cosets is called the quotient group of G by H and is denoted by G/H. The unit element of G/H is H and the inverse of a coset aH is the coset a -1H.

Homomorphic mappings of groups. Th following theorem is fundamental to the entire theory of homomorphic mappings:

Theorem. Under homomorphic mapping of an arbitrary group G onto a group G', the set N of elements of G that are mapped into the identity element e' of G' is a normal subgroup of G; the set of elements of G that are mapped into an arbitrary fixed element of G' is a coset of G with respect to N, and the one-to-one correspondence thus established between the cosets of G with respect to N and the elements of G' is an isomorphism between G' and the quotient group G/N.

Mathematics, Its Content, Methods and Meaning, III, p. 304

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References

James and James. Mathematics Dictionary.

Ayres. Modern Algebra. Chap. 9

Birkhoff, MacLane, A Survey of Modern Algebra. Chap. vi

Beaumont, Ball. Intro. to Modern Algebra and Matrix Theory. Chap. IV

Van der Waerden. Modern Algebra. Chap. 2

Mathematics, Its Content, Methods and Meaning. Chap. XX

Fang. Abstract Algebra.