Page 141 - DMTH403_ABSTRACT_ALGEBRA

Abstract Algebra Richa Nandra, Lovely Professional University

Notes Unit 13: Solvable Groups

CONTENTS
Objectives

Introduction
13.1 Solvable Group
13.2 Summary
13.3 Keywords
13.4 Review Questions

13.5 Further Readings

Objectives

After studying this unit, you will be able to:

Discuss the solvable groups

Describe examples of solvable group

Introduction

In the earlier unit, you have studied about the conjugate elements and Sylow’s Theorem. This
unit will equip you with more information related to solvable group.
13.1 Solvable Group

Definition: The group G is said to be solvable if there exists a finite chain of subgroups
G = N N ··· N such that
0
n
1
(i) N is a normal subgroup in N for i = 1, 2, ... ,n,
i-1
i
(ii) N / N is abelian for i = 1, 2, ..., n, and
i-1
i
(iii) N = {e}.
n
Proposition: A finite group G is solvable if and only if there exists a finite chain of subgroups
G = N N ... N such that
1
n
0
(i) N is a normal subgroup in N for i = 1, 2, . . ., n,
i-1
i
(ii) N / N is cyclic of prime order for i = 1, 2, . . ., n, and
i
i-1
(iii) N = {e}.
n
Theorem 1: Let p be a prime number. Any finite p-group is solvable.
Definition: Let G be a group. An element g in G is called a commutator if
g = aba b
-1 -1
for elements a, b in G.

134 LOVELY PROFESSIONAL UNIVERSITY