I know we can have left and right cosets without having the group be normal, but can we mod out by a subgroup without it being normal and still get a group?
Thank you in advance!
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1No. A subgroup is normal if and only if the quotient is a group. The quotient still makes sense when the subgroup is not normal, but it has no operation. That is indeed the purpose of singling out normal subgroups. – Matt Samuel Jan 01 '19 at 18:13
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Thanks! Could you give me an example, I'm having a hard time thinking of one – Dank Math Jan 01 '19 at 18:17
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Take $S_3$ and the subgroup of order $2$ generated by $(12)$. – Matt Samuel Jan 01 '19 at 18:18
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Search the site for the dupe answer. – Randall Jan 01 '19 at 18:18
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See https://math.stackexchange.com/questions/14282/why-do-we-define-quotient-groups-for-normal-subgroups-only/ – Tobias Kildetoft Jan 01 '19 at 18:19