Let M and N be normal subgroups of G and let N be a subgroup of M. Show that N i
ID: 3082717 • Letter: L
Question
Let M and N be normal subgroups of G and let N be a subgroup of M. Show that N is normal in M.Explanation / Answer
explicitly, let ?:G--> G/M x G/N be the map: ?(g) = (gM,gN). i claim that ?(G) is isomorphic to G. that means showing 3 things: 1. ? is onto ?(G), this is self-evident. 2. ? is 1-1. 3. ? is a homomorphism. let's tackle (3) first: ?(gh) = (ghM, ghN) = (gM,gN)(hM,hN) = ?(g)?(h). so ? is a homomorphism. it just remains to show that ? is 1-1. to do that, all we have to do is show that ker(?) = {e}. if g is in the kernel of ?, then (gM,gN) = (M,N), which means in turn, that g is in M, and g is in N. thus g is in M?N, hence g = e. thus ker(?) = {e}, so ? is an isomorphism of G with ?(G).
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