   # For each a in the group G , define a mapping t a : G → G by t a ( x ) = a x a − 1 . Prove that t a is an automorphism of G . Sec. 4.6 , # 32 ≪ Let a be a fixed element of the group G . According to Exercise 20 of Section 3.5 , the mapping t a : G → G defined by t a ( x ) = a x a − 1 is an automorphism of G . Each of these automorphisms t a is called an inner automorphism of G . Prove that the set I n n ( G ) = { t a | a ∈ G } forms a normal subgroup of the group of all automorphisms of G . ### Elements Of Modern Algebra

8th Edition
Gilbert + 2 others
Publisher: Cengage Learning,
ISBN: 9781285463230

#### Solutions

Chapter
Section ### Elements Of Modern Algebra

8th Edition
Gilbert + 2 others
Publisher: Cengage Learning,
ISBN: 9781285463230
Chapter 3.5, Problem 20E
Textbook Problem
5 views

## For each a in the group G , define a mapping t a : G → G by t a ( x ) = a x a − 1 . Prove that t a is an automorphism of G .Sec. 4.6 , # 32 ≪ Let a be a fixed element of the group G . According to Exercise 20 of Section 3.5 , the mapping t a : G → G defined by t a ( x ) = a x a − 1 is an automorphism of G . Each of these automorphisms t a is called an inner automorphism of G . Prove that the set I n n ( G ) = { t a | a ∈ G } forms a normal subgroup of the group of all automorphisms of G .

To determine

To prove: For each a in the group G, define a mapping ta:GG by ta(x)=axa1 is an automorphism of G.

### Explanation of Solution

Given information:

For each a in the group G, the mapping ta:GG defined by ta(x)=axa1.

Formula used:

Definition of automorphism:

The mapping ϕ:GG is an automorphism from G to G if

1. ϕ is a one-to-one correspondence (bijective).

2. ϕ is operation preserving (homomorphism).

Explanation:

The mapping ta:GG defined by ta(x)=axa1.

Let ta(x)=ta(y);x,yG.

By using the mapping ta,

axa1=aya1

Cancellation law holds when G is group a,a1,x,yG.

By using the cancellation law,

x=y

This proves that mapping ta is one-one mapping.

Let x be any element in group G.

Since G is closed under multiplication, there exists an element a1xaG such that

ta(a1xa)=a(a1xa)a1=(aa1)(x)(aa1)=e(x)e=xG

This shows that ta is onto mapping

### Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

#### The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

Find more solutions based on key concepts
f(x) = 3x3 x2 + 10; limxf(x); and limxf(x) x 1 5 10 100 1000 f(x) x 1 5 10 100 1000 f(x)

Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach

Add the following signed number: (6)+(+9)

Elementary Technical Mathematics

Finding a Derivative In Exercises 81-86, find F'(x). F(x)=2x21t3dt

Calculus: Early Transcendental Functions (MindTap Course List)

True or False: The figure at the right is the graph of the direction field for y = (y 1)x2.

Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th 