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### Abstract Algebra: An Introduction

Book edition 3rd
Author(s) Thomas W Hungerford, David Leep
Pages 608 pages
ISBN 9781111569624

# Let N be a normal subgroup of G , ${\mathbit{a}}{\mathbf{\in }}{\mathbit{G}}$ , and C the conjugacy class of a in G .If ${{\mathbit{C}}}_{{\mathbf{i}}}$ is any conjugacy class in G , prove that ${{\mathbit{C}}}_{{\mathbf{i}}}{\mathbf{\subseteq }}{\mathbit{N}}$ or ${{\mathbit{C}}}_{{\mathbf{i}}}{\mathbf{\cap }}{\mathbit{N}}{\mathbf{=}}{\mathbit{\Phi }}$ .

It is proved that, ${C}_{i}\subseteq N$ or ${C}_{i}\cap N=\Phi$ .

See the step by step solution

## Step 1: Given information

It is given that N is a normal subgroup of G , $a\in G$ , C is the conjugacy class of a in G and ${C}_{i}$ is any conjugacy class in G .

## Step 2: Prove the statement

If ${C}_{i}\cap N=\Phi$, then there is nothing to prove.

If ${C}_{i}\cap N$ is non-empty, then part (a) implies that ${C}_{i}\subseteq N$ .

It has been concluded that, ${C}_{i}\subseteq N$ or ${C}_{i}\cap N=\Phi$.