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Math Help - Isomorphic groups

  1. #1
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    Isomorphic groups

    Here is a set of four groups: Z4, Z2 x Z2, P2, V
    Determine which groups are isomorphic to which others.

    * P2 denotes the group of subsets of a two element set
    * V denotes the group of the four complex numbers {i, -i, 1, -1} with respect to multiplication.

    Thanks for the help!!
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by jzellt View Post
    Here is a set of four groups: Z4, Z2 x Z2, P2, V
    Determine which groups are isomorphic to which others.

    * P2 denotes the group of subsets of a two element set
    * V denotes the group of the four complex numbers {i, -i, 1, -1} with respect to multiplication.

    Thanks for the help!!
    How much group theory do you know? Since 4=2^2 all groups of order four are abelian. Then, by the structure theorem any group of order four is isomorphic to \mathbb{Z}_4 or \mathbb{Z}_2\oplus\mathbb{Z}_2 according to whether their is an element of order 4 or not.
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    Quote Originally Posted by jzellt View Post
    Here is a set of four groups: Z4, Z2 x Z2, P2, V
    Determine which groups are isomorphic to which others.

    * P2 denotes the group of subsets of a two element set
    * V denotes the group of the four complex numbers {i, -i, 1, -1} with respect to multiplication.

    Thanks for the help!!


    Well, the first group is the cyclic one of order 4, and the second one is the Klein group (non-cyclic

    of order 4), so what kind of groups of order 4 are the 3rd and 4th ones?

    Tonio
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