Math Is Fun Forum

thandu3

Member

Registered: 2008-10-21

Posts: 4

isomorphism of group problem

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?
email: Email removed to protect from spammers- Ricky

i have wrote intersection,normal group literary rather than using sign.H1,H2,..........Hn

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: isomorphism of group problem

Number 1 is proving that HN form a group.  This is just like any problem proving that something is a group.  But remember that we know HN is a subset of G (why?).  Some properties of groups come immediately if we know that we are talking about a subset of a group.  Which properties come immediately?

For the 2nd part of this one, we must show that N is normal in HN.  How do we show that something is normal (in other words, what's the definition)?

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."