 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