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Richa Nandra, Lovely Professional University Unit 11: Conjugate Elements

Unit 11: Conjugate Elements Notes

CONTENTS
Objectives

Introduction
11.1 Conjugate Subgroup
11.2 Summary
11.3 Keywords
11.4 Review Questions

11.5 Further Readings

Objectives

After studying this unit, you will be able to:

Define conjugate subgroup

Discuss conjugacy class of an element

Introduction

In the last unit, you have studied about finite abelian group. If G is a group and X is an arbitrary
set, a group action of an element g  G and x  X is a product, g giving in x many problem in
x
algebra may best be attached in group actions. In this unit, you will get the information related
to conjugate elements.
11.1 Conjugate Subgroup

Definition: Let G be a group, and let x, y be elements of G. Then y is said to be a conjugate of x
if there exists an element a in G such that y = axa .
-1
If H and K are subgroups of G, then K is said to be a conjugate subgroup of H if there exists an
element a in G such that K = aHa .
-1
Proposition 1:
(a) Conjugacy of elements defines an equivalence relation on any group G.

(b) Conjugacy of subgroups defines an equivalence relation on the set of all subgroups of G.
Definition: Let G be a group. For any element x in G, the set
{ a in G | axa = x }
-1
is called the centralizer of x in G, denoted by C(x).
For any subgroup H of G, the set
{ a in G | aHa = H }
-1
is called the normalizer of H in G, denoted by N(H).

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