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#1 2007-01-24 12:51:09

Stanley_Marsh
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Registered: 2006-12-13
Posts: 345

A newbie question

I don't understand why the intersection of groups is still a group ?


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#2 2007-01-24 13:18:27

Stanley_Marsh
Member
Registered: 2006-12-13
Posts: 345

Re: A newbie question

here is the question Prove that the intersection of a family of subgroups of a group G is a subgroup of G.


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#3 2007-01-24 13:20:22

Stanley_Marsh
Member
Registered: 2006-12-13
Posts: 345

Re: A newbie question

and another one , Let a,b,c,d, be integers such that ad-bc=1 , for integer u, v define
m=au+bv , n=cu+dv .  Prove that (m,n) = (u,v)

Last edited by Stanley_Marsh (2007-01-24 13:20:38)


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#4 2007-01-25 04:33:27

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

Re: A newbie question

Yes, this can be a bit confusing at first.  Doesn't seem like it should be true, right?  But, just run through all the properties:

Let A and B be two subgroups of G.  We let C signify A intersect B.

e must be in A, and e must be in B, so e must be in C.  Thus, C contains the identity.

Now we let some two elements x and y be in C.  Then x is in A and x is in B and y is in A and y is in B.  So xy is in A as A must be closed, and for the same reason, xy is in B.  So xy is in C, and C is closed.

Now let x be in C.  So x is in A and x is in B.  Since A and B are closed to inverses, x inverse is in A and x inverse is in B.  So x inverse is in C, and thus, C is closed ot invserses.

Finally, we can just say that C is a subset of G, and thus, all elements in C must be associative, as all elements in G are.

And that's it.  Also note that we can expand C to be the intersection of A_1, A_2, A_3, ..., A_n, and all the above statements still apply.


"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..."

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#5 2007-01-26 10:07:44

Stanley_Marsh
Member
Registered: 2006-12-13
Posts: 345

Re: A newbie question

Oh ,yeah, I can assume x, y in C ,  instead I assume them in other wrong places , It's really confusing at the beginning ~


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