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 anelement g e G normalizes N if and only if gSg¯1 c N.

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

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Related questions

Question

100%

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 $g S g^{- 1} \subset N$

Trending now

This is a popular solution!

Step by step

Solved in 2 steps

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Factorization$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.

Similar questions

Recommended textbooks for you

Advanced Engineering Mathematics

Advanced Math

ISBN:

9780470458365

Author:

Erwin Kreyszig

Publisher:

Wiley, John & Sons, Incorporated

Numerical Methods for Engineers

Advanced Math

ISBN:

9780073397924

Author:

Steven C. Chapra Dr., Raymond P. Canale

Publisher:

McGraw-Hill Education

Introductory Mathematics for Engineering Applicat…

Advanced Math

ISBN:

9781118141809

Author:

Nathan Klingbeil

Publisher:

WILEY

Advanced Engineering Mathematics

Advanced Math

ISBN:

9780470458365

Author:

Erwin Kreyszig

Publisher:

Wiley, John & Sons, Incorporated

Numerical Methods for Engineers

Advanced Math

ISBN:

9780073397924

Author:

Steven C. Chapra Dr., Raymond P. Canale

Publisher:

McGraw-Hill Education

Introductory Mathematics for Engineering Applicat…

Advanced Math

ISBN:

9781118141809

Author:

Nathan Klingbeil

Publisher:

WILEY

Mathematics For Machine Technology

Advanced Math

ISBN:

9781337798310

Author:

Peterson, John.

Publisher:

Cengage Learning,

Basic Technical Mathematics

Advanced Math

ISBN:

9780134437705

Author:

Washington

Publisher:

PEARSON

Topology

Advanced Math

ISBN:

9780134689517

Author:

Munkres, James R.

Publisher:

Pearson,

SEE MORE TEXTBOOKS

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.