This specifically relates more towards the argument as to why an inverse exists.(adsbygoogle = window.adsbygoogle || []).push({});

First the problem

The normalizer is defined as follows, N_{G}(H)={g^{-1}Hg=H} for some g in N_{G}(H). I get why identity exists and why the operation is closed. It is in arguing that an inverse exists that I have beef. Specifically this argument:

eHe= (g^{-1})^{-1}g^{-1}Hgg^{-1}=(g^{-1})^{-1}Hg^{-1}

so g^{-1}is in N_{G}(H)

This above proof I found in the book Groups, rings, and fields by D.A.R. Wallace.

I wondering why this is more valid than this:

H=gHg^{-1}

g^{-1}H=Hg^{-1}

g^{-1}Hg=H

So g^{-1}is in N_{G}(H)

I'm sorry if this question is so dense someone has an aneurysm

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Normalizer subgroup proof proving the inverse

**Physics Forums | Science Articles, Homework Help, Discussion**