Answered: 28) Let N be a finite subgroup of a…

28) Let N be a finite subgroup of a group G and assume N =< S > for some subset S of G. Prove that an element g E G normalizes N if and only if gSg-1 cN.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter4: More On Groups

Section4.6: Quotient Groups

Problem 19E

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abstract algebra question

28) Let N be a finite subgroup of a group G and assume N =< S > for some subset S of G. Prove that an
element g e G normalizes N if and only if gSg¯1 c N.

Expert Solution

Step 1

Given that $N$ is a finite subgroup of a group $G$ and suppose $N = <S>$ for some subset $S$ of $G$

To prove that an element $g \in G$ normalizes $N$ if and only if $g S g^{- 1} \subset N$

Now, $N$ is a finite subgroup of a group $G$ generated by $S$ which is a subset of $G$ .

Therefore, $N$ can be written as $N = \{s_{1}, s_{2}, s_{3}, . . ., s_{n}\}$ where $s_{k}$ is an element of $S$ for k=1,2,3,...,n

Thus, $S$ is contained in $N$ i.e. $S \subset N$ .

Now, let us consider that an element $g \in G$ normalizes $N$

To prove that $g S g^{- 1} \subset N$

Since $g$ normalizes $N$ , therefore, for all $x \in N, g x g^{- 1} \in N$

Let $y \in S$ be an arbitrary element. Since $S \subset N$ , therefore, $g y g^{- 1} \in N$

As y is chosen arbitrarily, for all $y \in S$ , $g y g^{- 1} \in N$ which implies $g S g^{- 1} \subset N$

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