Second Isomorphism Theorem
Intuition on the Second Isomorphism Theorem from Group Theory
The quotient group G/N restricted to cosets generated by a subgroup H<G has group structure isomorphic to the quotient of the individual restrictions HnG and HnN. ie (G/N)"n"H === HnG/HnN = H/HnN.
‘Restricting’ and ‘quotienting’ commute via this isomorphism. That’s the logos of what the second isomorphism theorem is about.
I used to see the second isomorphism theorem as an ugly duckling compared to the masterpiece that is the first isomorphism theorem, and the notationally-clear third isomorphism theorem, because at first glance the H/HnN === HN/N notation doesn’t convey anything meaningful to me. But now I appreciate it.
This video also highlights a subtlety: just because H is a proper subgroup of G does not necessarily mean that HN/N is going to be a proper subgroup of G/N. For example with G=Z, H=6Z, N=7Z, we have HN=1Z, HnN=42Z, 6Z/42Z = H/HnN === HN/N = 1Z/7Z = G/N, because HN spans the entirety of G.