Results 1 to 3 of 3

Math Help - Which Group is this Group Isomorphic to?

  1. #1
    Member
    Joined
    Oct 2009
    Posts
    128

    Which Group is this Group Isomorphic to?

    Hello!

    Using the external direct product \oplus, the group
    S_3 \oplus Z_2 is isomorphic to one (and only one!) of the following groups (this question reminds me of a game show...)

    a) Z_{12}
    b) Z_6 \oplus Z_2
    c) A_4
    d) D_6

    I am assuming these groups are the well known groups
    Z_n are the integers modulo n under addittion
    A_n is the alternating group (all even permutations of n) under function composition
    D_n is the dihedral group of order 2n
    and S_n is the set of permutations of n

    The question recommends determining which groups it is not isomorphic to and determining the answer by elimination.

    I started by the question noting that S_3 \oplus Z_2 is cyclic, I believe? so it must be isomorphic to a cyclic group as well...
    and also it must have the same number of elements of each order of any group it is isomorphic... but even armed with those facts I am having trouble finding that any of a), b), c) or d) are isomorphic to the given group S_3 \oplus Z_2

    Any help appreciated!
    Thank you!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member roninpro's Avatar
    Joined
    Nov 2009
    Posts
    485
    Actually, S_3\oplus \mathbb{Z}_2 cannot be cyclic. It isn't even abelian! With this in mind, you can remove (a) and (b) from consideration.

    To deal with (c) and (d), I would try looking at the orders at some elements in each group. Compare them with your S_3\oplus \mathbb{Z}_2.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21
    Quote Originally Posted by roninpro View Post
    Actually, S_3\oplus \mathbb{Z}_2 cannot be cyclic. It isn't even abelian! With this in mind, you can remove (a) and (b) from consideration.

    To deal with (c) and (d), I would try looking at the orders at some elements in each group. Compare them with your S_3\oplus \mathbb{Z}_2.
    Alternatively you can show that S_3\oplus\mathbb{Z}_2\not\cong A_4. To see this note that S_3\oplus\{0\}\leqslant S_3\oplus\mathbb{Z}_2 and \left|S_3\oplus\{0\}\right|=6 but A_4 has no subgroup of order 6.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Determine if group / isomorphic
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: September 26th 2011, 12:45 PM
  2. isomorphic group
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: May 5th 2011, 10:51 PM
  3. Any group of 3 elements is isomorphic to Z3
    Posted in the Advanced Algebra Forum
    Replies: 11
    Last Post: November 2nd 2010, 06:30 AM
  4. Prove a group is isomorphic to another
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: February 8th 2010, 04:19 PM
  5. Prove a group H is isomorphic to Z.
    Posted in the Advanced Algebra Forum
    Replies: 5
    Last Post: September 23rd 2009, 11:08 PM

Search Tags


/mathhelpforum @mathhelpforum