G is a group and N a subgroup of G, and if index of N in G is 2, then N is Normal. Can someone please explain why?
Group Theory : How is N Normal?
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group-theory
normal-subgroups
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0Any attempts of yours? – 2017-02-04
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0this question is a duplicate to: http://math.stackexchange.com/questions/84632/subgroup-of-index-2-is-normal and http://math.stackexchange.com/questions/420583/normal-subgroups-and-factor-groups – 2017-02-04
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0Thanks a lot. I searched for quite a while but I couldn't find it. – 2017-02-04
1 Answers
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Suppose $N$ such that $[G:N]=2$. Thus $N$ has two left cosets (and two right cosets) in $G$. If $g \in N$, then $gN = N = Ng$. If $g \notin N$, then $gN=G∖N$ as there are only two cosets and the cosets partition $G$.
For the same reason, $g \notin N \implies Ng = G∖N$.
That is, $gN=Ng$. Thus $N$ is normal in $G$.