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Subgroup of G

  1. Sep 21, 2008 #1
    1. The problem statement, all variables and given/known data
    Let G be a group such that every x in G\{e} has order 2.
    (a) Let H<=G be a subgroup. Show that for every x in G the subset H U (xH) is also a subgroup.
    (b) Show that if G is finite, then |G|=2^n for some integer n.

    3. The attempt at a solution
    For (a), I know that since x is also part of G, then if multiplied to H (left coset) it will also still be contained in G, but I don't know how to prove it.
    For (b), since the order of every element in G is 2, then every set in G has 2 elements, so the order of G is just 2(elements) times how many sets there are...also having trouble proving this.

    Please let me know if my train of thoughts are right.
  2. jcsd
  3. Sep 21, 2008 #2


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    Hint for (a): Notice that every element is its own inverse.

    Hint for (b): Use part (a).
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