Isomorphisms and Automorphisms Groups

Definition :- A group is a non-empty set G , together with a binary composition (operation) * on G , such that it satisfies the following postulates. (i) Closure property for all a , b Є G = a * b Є G (ii) Associativity : a*(b*c) = ( a * b ) * c , for all a , b , c Є G (iii) Existence of identity : There exist an element e Є G such that e * a = a * e = a for all a Є G (iv) Existence of inverse : for each a Є G , there exists an element a Є G such that a * a' = a' * a = e Here a' is called inverse of a and a' = a-1 Since * is a binary operation on G , a * b Є G for all a, b Є G

1. Generally we denote the binary composition for a group by. (dot). Note that a group is not just a set G. Infact a group G is made up of two entities. The set G and a binary operation. on G. 2. The symbol * and . are just notations representing binary operation. Defination :- A group G is obtaine if its binary operation is commutative i.e. a . b = b . a for all a , b Є G. Some Results Based on Groups. In a group < G , . > (i) Identity element is unique. (ii) Inverse of each a Є G is unique. (iii) a-1 is called inverse of a and (a-1) -1 = a for all a Є G (iv) (ab)-1 = b-1 a-1 , for all a , b Є G (v) ab = ac => b=c (left cancellation Law) ba = ca => b = c (Right cancellation Law) (vi) ea = a (left identity) ae =a ( Right identity ) (vii) a-1 a = e (left inverse) (viii) aa-1 = e (Right inverse) (ix) A system < G , . > forms a group iff (a) a (bc ) = (ab ) c for all a , b , c Є G (b) There exist e Є G such that ae = a for all a Є G (c) For all a Є G , there exist a' Є G such that aa' = e on G. If G is a group with identity element e, then < G, . > and < {e},. > are called trivial subgroups of < G,. > Any subgroup other than these two subgroups is called a proper or non- trivial subgroup. Definition:- A subgroup H of a group G is said to be a normal subgroup of G if Ha = aH for all a Є G G and { e } are always normal subgroups of G and these group are called trivial normal subgroups . A group G = { e } not having any non-trial normal subgroups is called a simple group , e.g. H = { 1 , -1 } is a normal. Definition:- Let < G , 0 > and < G’ * > be two groups . Then a mapping f: G → G' is called a homomorphism, then f (G) is called homomorphism image of G. A one to one homomorphism is called a monomorphism. A homomorphism of a group G to itself is called an endomorphism of G. An onto homomorphism is called an epimorphism. Some Results Based on Homomorphisms. (i) Let f : G→ G' be a homomorphism. Then

(b) f ( a-1 ) = ( f (a) )-1

(c) f (an ) = [ f (a) ]n , n is an integer , where e and e' are identity elements of G and G' respectively. Definition :- A homomorphism of a group G onto a group G' is called an isomorphism if f is a one to one mapping. Definition :- An isomorphism from a group G to itself is called an automorphism of G . Thus a one-one onto map f : < G , * >→ is called an automorphism of G if f ( x *y ) = f (x) * f (y) for all x , y Є G . Thm :- Let G be a group and let Aut (G) denot the set of all automorphism of a group G . Then Aut (G) forms a group under the composition of mapping as binary operation. Proof :- Let Aut (G) = {f : f is an automorphism of G} We shall show that Aut (G) forms a group with composition of mapping as binary operation. If x ,y Є G then (gf ) (xy ) = g [f (xy )] = g [ f (x) f (y) ] = g [ f(x) ] [g f(y) ] = (gf ) (x) (gf) (y) Thus gf is also an automorphism of G and so gf Є Aut (G). This shows that automorpisms holds closure. Associativity :- Let f, g, h Є Aut G ( f (gh ) (x) = f ( gh (x) ) = f (g (h (x) ) --------1. ((fg)h ) (x) = (fg ) h (x) = f(g(h(x)))---------2. From eq. 1 and 2 (f(gh)(x) = ((fg)h)(x) , for all x Є G f(gh) = (fg) h for all f , g, h Є Aut (G) Existence of Identity: Let I: G →G be the identity function on G, such that I (x) = x for all x Є G So I is one-one onto Let x ,y Є G => xy Є G = > I (xy ) = xy = I (x) I (y) => I (xy) = I (x) I (y) => I is a homomorphism. Thus I : G → G is an isomorphism of G onto itself and so I Є Aut (G) Existence of Inverse :- Let f Є Aut (G) => f is one-one onto mapping form G to G = > f -1 is also one-one onto mapping form G to G. Let x ,y Є G Then there exist x , y Є G , such that f -1 (x) = x 1 = f (x1) = x f-1 (y) = y , = f (y1) = y Then f-1 (xy) = f-1 [f (x1) f (y1) ] = f-1 (f (x , y ) ) ( f is homomorphism ) = x1 y1 = f-1 (x) f-1 (y)

Hence Aut (G) forms a group with respect to composite composition . Definition:- Let G be a group and a Є G . Then the mapping Ta : G → G defined by Ta (x) : a-1 xa is an automorphism of G and it is called an inner automorphism of additive group of integers Ta (x) = (-a) + x + (a) = x for all x Є G . Remarks:- Performing Ta on x Є G is called conjugation of x by a . 2. We denote the set of all inner automorphisms of G by Inn (G) Value Addition :- If G is an abelian group then the inner automorphism induced by a Є G reduces to the identity automorphisms of G , i.e. Inn (G) = { I } . Thus, inner automorphisms are of interest mainly in case of non-abelian groups. Example:- We now study the inner automorphism of G = D4 the Dihedral group of symmetries of a square , induced by R90

R90 (R90) = R90 R90 R90-1 = R 90 R90(R180) = R90R180 R90-1 = R180 R90 (R270) = R90 R270 R90-1 = R270 R90(H) = R90HR90-1 H R90 (V) = R90 V R90-1 = V R90 (D) = R90 D R90-1=D R90 (D') = R90 D' R90-1 = D'

It is good exercise to see the action of inner automorphism of D4 induced by all other elements of D4 as well. Problem: Let g be an element of a group G. show that the inner automorphism induced by g is same as the inner automorphism induced by Zg, Where z is in Z (G) , the centre of G Solution:- ɸzg (x) = (zg) x (zg)-1 as x Є G = g(z) x (g-1 z-1 ) , as z Є Z (G) = gz (x) z-1g-1 , as z Є Z ( G) = z-1 Є Z (G)

= gxzz-1g-1

Hence ɸzg = ɸg Note :- It is evident from the above problem that two distinct elements of a group G need not induce two distinct inner automorphisms , i.e. if a ≠ b in G even then a and b may be same. Thus if G = {e, a, b, c------- } then Inn (G ) = {ɸe , ɸb , ɸc………….} may have duplications. Ilustration :- Aut (Z6) Solutions : Aut (Z6 )≈ U(6) = { 1,5 } mod 6 Thus Aut (Z6) being a group of order Z (Prime ) is cyclic and hence isomorphic to Z2.

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Department of Mathematics, P.K.S.D. College, Kanina, Mohindergarh, Haryana, India E-Mail – vasant.r.deo@gmail.com