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Math Help - Groups

  1. #1
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    Groups, subgroups prove

    Let (G,*) is group and A,B are her subgroups.
    - Prove that AB is subgroup of (G,*).
    - If |A|=10 and |B| =7 how many elements AB have and who are they?
    - Find subgroups A,B from (Z,+) so that A
    ∪B isn't subgroup of (Z,+).
    thanks
    Last edited by kljoki; August 16th 2012 at 11:53 AM.
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  2. #2
    MHF Contributor

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    Re: Groups

    how do you prove a subSET is a subGROUP?

    1) you verfiy the subset satisfies the group axioms:

    a) closure (if a,b are in a subset S of G, we need a*b to be in S as well).
    b) associativity (but this is obvious...why?)
    c) existence of an identity (hint: doesn't this have to be the same identity as the one for G?)
    d) existence of inverses

    to save some time: prove that (a) & (d) together imply (c). this (together with (b) being obvious), only gives you two things to check, instead of four.

    2) alternative method: show that for all a,b in S, a*b-1 is also in S. why is this just as good as doing (1)?

    (hint: consider a = b, first, to show that e is in S. next, consider a = e, to show that b-1 is in S. finally, use the fact that b = (b-1)-1, to show that ab is in S).

    to continue, observe that x in A⋂B, means: x is in A, AND x is in B.

    to answer the next part, you need to know that |A⋂B| divides |A| and |B| (lagrange's theorem). what is gcd(10,7)?

    for the last part: consider 2Z U 3Z. is 2+3 = 5 in this?
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