Let G be a group. Given an element a of G, define the function La : G → G by La(x)=ax. (We will call this function “left multiplication by a.”) Define Ra : G → G by Ra(x) = xa. (This is “right multiplication by a”.) (a) Show that La is a one-to-one function from G onto G (that is, a bijection from G to G.) (b) Show that for all a,b in G, LaLb =Lab and show that for all a,b in G, RaRb =Rba. (d) (Here SymG represents the set of all permutations on the set G. Some authors, eg. Dummit & Foote, write SG for this set of permutations.) Show that the map G → SymG defined by a ?→ La is an isomorphism from G into SymG.

Let G be a group. Given an element a of G, define the function La : G → G by La(x)=ax. (We will call this function “left multiplication by a.”) Define Ra : G → G by Ra(x) = xa. (This is “right multiplication by a”.) (a) Show that La is a one-to-one function from G onto G (that is, a bijection from G to G.) (b) Show that for all a,b in G, LaLb =Lab and show that for all a,b in G, RaRb =Rba. (d) (Here SymG represents the set of all permutations on the set G. Some authors, eg. Dummit & Foote, write SG for this set of permutations.) Show that the map G → SymG defined by a ?→ La is an isomorphism from G into SymG.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter3: Groups

Section3.5: Isomorphisms

Problem 20E: For each a in the group G, define a mapping ta:GG by ta(x)=axa1. Prove that ta is an automorphism of...

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