normal subgroup and cosets

**apple2009** · November 8th 2009, 05:11 PM

If H is a subgroup of a finite group G of index [G: H] two, then H is normal in G.
I need to show that the left cosets equals to the right cosets, but what are the cosets here?

**NonCommAlg** · November 8th 2009, 11:10 PM

Originally Posted by apple2009

If H is a subgroup of a finite group G of index [G: H] two, then H is normal in G.
I need to show that the left cosets equals to the right cosets, but what are the cosets here?

there are more than one way to prove this. one way is to write $G=H \cup xH,$ where $x \notin H.$ let $g \in G.$ if $g \in H,$ then clearly $g H g^{-1} = H.$ so we may assume that $g = xh,$ for some $h \in H.$

then $gHg^{-1}=xhHh^{-1}x^{-1}=xHx^{-1}.$ now if $xh_1x^{-1} \in xH,$ for some $h_1 \in H,$ then $x \in H,$ which is a false result. so $xHx^{-1} \subseteq H$ and we're done.

by the way, i just realized that you've already posted this question in here. next time don't post your question twice.

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