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).

Navigate

Let M = (Q, Sigma, delta, q_1, F) be a DFA recognizing A and p be the number of

Let M be a 6 times 6 matrix with columns [M_1,..., M_6]. Let N be an invertible

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.

Let M and N be normal subgroups of G and let N be a subgroup of M. Show that N i

Question

Explanation / Answer

Related Questions

Navigate