Given a group extension, wikipedia said that the group $N$ is isomorphic to a normal subgroup in $G$. However, how can I deduce it? Can't see the reason directly. Or am I misunderstand it?
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1It's the definition of a short exact sequence for groups. You need that condition so that the quotient makes sense as well. – Alvaro Martinez Apr 13 '19 at 16:00
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1$N$ is the kernel of the homomorphism $G\to Q$. – lulu Apr 13 '19 at 16:01
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I see! The kernel of a homomorphism must be normal. – Eric Apr 13 '19 at 16:02
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@lulu Wait... Why $N$ is the kernel of $G\to Q$? Is it part of the definition of group extension? – Eric Apr 13 '19 at 16:18
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1That's part of the definition of an exact sequence. – lulu Apr 13 '19 at 16:22
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This may also help with some intuition about it: https://math.stackexchange.com/questions/776039/intuition-behind-normal-subgroups/3732426#3732426 – Ciro Santilli OurBigBook.com Aug 16 '20 at 07:54