Is it true that for a Group G with Normal Group N:GN=GNN?I think the statement is correct. But why do we have to write: [G,G]NN here instead of just G,GN?

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amarantha41 · Accepted Answer

(1) N is normal subgroup of G; so it is normal subgroup of every subgroup between N and G.   (2) When talking of (some subgroup)/N, this (some subgroup) should be taken containing N.   (3) We ask: given a normal subgroup N of G, does it always contain [G,G]? You can easily see negative answer by taking simple examples of finite order groups G.   (4) Thus, when we want to factor [G,G] by N, first look smallest subgroup above N(=containing N) which contains [G,G]; it is precisely [G,G]N.

Becky Harrison · Accepted Answer

[G,G]NN is the image of [G,G] under the canonical homomorphism from G to GN. This homomorphism respects taking commutators. So in particular [GN,GN]=[G,G]NN.

Is it true that for a Group G with Normal

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