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Problem 94 (The Normal Subgroup Test)

Suppose that $G$ is a group and $H$ is a subgroup of $G$. Prove that $H$ is a normal subgroup of $G$ if and only if $xHx^{-1}\subseteq H$ for all $x\in G$.

We already know that $H$ is normal if and only if $xHx^{-1}=H$ for all $x\in G$, so we know $H$ is normal implies $xHx^{-1}\subseteq H$ for all $x\in G$. What you must show is that if $xHx^{-1}\subseteq H$ for all $x\in G$, then $H$ must be normal (so $Hx=xH$ for all $x\in G$).

The following pages link to this page.

Problem.HomomorphismsPreserveNormalSubgroups
Problem.TheNormalSubgroupTest
Schedule.20161128
Schedule.20161130
Schedule.20171129
Schedule.AllProblems

Solution.TheNormalSubgroupTestJosh

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Page last modified on November 27, 2017, at 07:40 AM

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HomePage Current Problems All Problems Pictures ProblemsByWeek List of Problems List of Definitions List of Theorems List of Conjectures All Recent changes (use this to find a page that may have been lost)	Problem The Normal Subgroup Test Please Login to access more options. Problem 94 (The Normal Subgroup Test) Suppose that $G$ is a group and $H$ is a subgroup of $G$. Prove that $H$ is a normal subgroup of $G$ if and only if $xHx^{-1}\subseteq H$ for all $x\in G$. We already know that $H$ is normal if and only if $xHx^{-1}=H$ for all $x\in G$, so we know $H$ is normal implies $xHx^{-1}\subseteq H$ for all $x\in G$. What you must show is that if $xHx^{-1}\subseteq H$ for all $x\in G$, then $H$ must be normal (so $Hx=xH$ for all $x\in G$). The following pages link to this page. Problem.HomomorphismsPreserveNormalSubgroups Problem.TheNormalSubgroupTest Schedule.20161128 Schedule.20161130 Schedule.20171129 Schedule.AllProblems Solution.TheNormalSubgroupTestJosh Edit Page History Source Attach File Backlinks List Group Page last modified on November 27, 2017, at 07:40 AM
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