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Math Help - Group theory question with intersections, dividing, gcd...

  1. #1
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    Group theory question with intersections, dividing, gcd...

    Let H and K be subgroups of a finite group G. Let H∩K be a subgroup of G.

    (a) For any elements a and b of H, prove that a(H∩K) = b(H∩K) if and only if ak = bK.

    (b) Deduce from (a) that the number of elements in the set HK is: |H| |K| / |H∩K|.

    (c) If [G:H] and [G:K] have greatest common divisor 1, prove that G = HK and [G:H∩K] = [G:H] [G:K]

    Any help would be greatly appreciated!
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  2. #2
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    Re: Group theory question with intersections, dividing, gcd...

    (a) suppose for two elements of H, a(H∩K) = b(H∩K).

    what does this really mean? it means that b-1a is in H∩K, that is it lies in both H AND K.

    well, a and b are elements of H, and H is a subgroup, so b-1 is in H, and by closure b-1a is also in H. we knew that already.

    what is INTERESTING, is now we know b-1a is also in K.

    and this means that aK = bK.

    on the other hand suppose for two elements a and b of H, that aK = bK.

    then b-1a is in K. and as above, b-1a is certainly an element of H.

    so b-1a lies in both H AND in K, and so must be in H∩K.

    thus a(H∩K) = b(H∩K).

    (b) let's try to count the number of elements in HK = {hk: h in H, k in K}.

    well, we obviously have |H| choices for h, and |K| choices for k.

    so we have "at most" |H|*|K| elements of HK.

    the trouble is, what if we have "duplicates", that is: hk = h'k'? we don't want to count those "more than once".

    for example, in the group (Z30,+), we have the subgroup H = <4> and K = <6>.

    now 10 is in HK, since 10 = 4 + 6. but we also have: 10 = 16 + 24 = 4(4) + 4(6), which is also in HK (4+4+4+4 is in H, and 6+6+6+6 is in K).

    so the sums 4+6 and 16+24 both yield the same answer, 10. we only want to count 10 as an element of HK ONCE.

    so what does hk = h'k' tell us, in terms of cosets? it tells us:

    h = h(k'k-1). but k'k-1 is clearly in K, so h is in h'K, so hK = h'K.

    as we saw above, this means h(H∩K) = h'(H∩K). so hk = h'k' only if h and h' are in the same coset of (H∩K).

    now let's look at it from the other end, suppose h(H∩K) = h'(H∩K). this means that hK = h'K, so

    every time h and h' are in the same coset of H∩K, we get hK = h'K, so for any k in K, hk is a duplicate of some h'k', with k' in K.

    that is: hk = h'k' if and only if h(H∩K) = h'(H∩K). so we can't multiply |H| by |K|, we have to factor out the |H∩K| duplicates

    (in other words we don't have |K| multiples of each h, we only have [K:H∩K] multiples - we get distinct elements hk, for each h in H, and each COSET k(H∩K)).

    so |HK| = |H|*[K:H∩K] = |H|*(K/|H∩K|) = (|H|*|K|)/|H∩K|.

    (c) this should be easy, now.
    Last edited by Deveno; November 7th 2012 at 05:32 PM.
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