Page 135 - DMTH403_ABSTRACT_ALGEBRA

Abstract Algebra Richa Nandra, Lovely Professional University

Notes Unit 12: Sylow’s Theorems

CONTENTS
Objectives

Introduction
12.1 The Sylow Theorems
12.1.1 A Proof of Sylow’s Theorems
12.2 Summary
12.3 Keywords

12.4 Review Questions
12.5 Further Readings

Objectives

After studying this unit, you will be able to:
Discuss Sylow’s Theorem

Describe examples of Sylow’s Theorem

Introduction

We already know that the converse of Lagrange’s Theorem is false. If G is a group of order m and
n divides m, then G does not necessarily possess a subgroup of order n. For example, A has
4
order 12 but does not possess a subgroup of order 6. However, the Sylow Theorems do provide
a partial converse for Lagrange’s Theorem: in certain cases they guarantee us subgroups of
specific orders. These theorems yield a powerful set of tools for the classification of all finite
non-abelian groups.

12.1 The Sylow Theorems

We will use the idea of group actions to prove the Sylow Theorems. Recall for a moment what
it means for G to act on itself by conjugation and how conjugacy classes are distributed in the
group according to the class equation. A group G acts on itself by conjugation via the map (g, x)
 gxg . Let x ,...,x be representatives from each of the distinct conjugacy classes of G that consist
-1
1
k
of more than one element. Then the class equation can be written as
|G| = |Z(G)| + [G : C(x )] + ... + [G : C(x )],
1
k
where Z(G) = {g  G : gx = xg for all x  G} is the center of G and C(x ) = {g  G : gx = x g} is the
i
i
i
centralizer subgroup of x . i
We now begin our investigation of the Sylow Theorems by examining subgroups of order p,
where p is prime. A group G is a p-group if every element in G has as its order a power of p,
where p is a prime number. A subgroup of a group G is a p-subgroup if it is a p-group.

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