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# TIFR 2013 Problem 10 Solutions - Normal Subgroup of Order 2 TIFR 2013 Problem 10 Solutions is a part of TIFR entrance preparation series. The Tata Institute of Fundamental Research is India's premier institution for advanced research in Mathematics. The Institute runs a graduate programme leading to the award of Ph.D., Integrated M.Sc.-Ph.D. as well as M.Sc. degree in certain subjects.
The image is a front cover of a book named Contemporary Abstract Algebra by Joseph A. Gallian. This book is very useful for the preparation of TIFR Entrance.

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## Problem:True/False?

Any normal subgroup of order 2 is contained in the center of the group.

## Discussion:

If $N$ is a normal subgroup of a group $G$ and $|N|=2$, then $N=\left\{e,a\right \}$ where $a^2=e$.

For all $g\in G$ we have $gag^{-1}\in N$.

Can $gag^{-1}=e$? No. Since that would imply $a=e$.

Therefore, for all $g\in G$, $gag^{-1}=a$.

Which proves that a is in the center of the group.

## Helpdesk

• What is this topic:Abstract Algebra
• What are some of the associated concept: Normal Subgroup, Center of a Group
• Book Suggestions: Contemporary Abstract Algebra by Joseph A. Gallian

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