Definition: when H, K are subgroups of G, we define HK to be the set of all elements of G that can be written in the form hk where h is in H and k is in K.

1) let H be a subgroup of a group G and N be a normal subgroup of G.show that HN is a subgroup of G and N be a normal subgroup of HN.

2) let H,K and N be a subgroup of a group G, K is normal subgroup of H and N is normal subgroup of G.prove that NK is normal subgroup of NH.

3) let H1 and H2 be subgroups of a group G and N1 subgroup of H1 and N2 subgroup of H2.then show that

N1(H1 intersection N2) is normal subgroup of N1(H1 intersection H2)

and (H1 intersection N2)(H2 intersection N1) normal subgroup of (H1 intersection H2)

can u help me of these 3 proofs?

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