Results 1 to 2 of 2

Math Help - Groups of Order 8

  1. #1
    Newbie
    Joined
    Nov 2009
    Posts
    12

    Groups of Order 8

    Prove that no pair of the following groups of order 8: I8 (congruence class mod 8), I4 x I2, I2 x I2 x I2, D8 (dihedral group), Q are isomorphic.

    The link http://www.mathhelpforum.com/math-he...tml#post392182 shows how D8 and Q are not isomorphic, but I have no idea how to prove the rest of the pairs are not isomorphic.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Banned
    Joined
    Oct 2009
    Posts
    4,261
    Thanks
    2
    Quote Originally Posted by johnt4335 View Post
    Prove that no pair of the following groups of order 8: I8 (congruence class mod 8), I4 x I2, I2 x I2 x I2, D8 (dihedral group), Q are isomorphic.

    The link http://www.mathhelpforum.com/math-he...tml#post392182 shows how D8 and Q are not isomorphic, but I have no idea how to prove the rest of the pairs are not isomorphic.

    Well, the first 3 are abelian: \mathbb{Z}_8\,,\,\mathbb{Z}_4\times \mathbb{Z}_2\,,\,\mathbb{Z}_2\times \mathbb{Z}_2\times \mathbb{Z}_2 , so they can't be isomorphic with the non-abelian quaternion and dihedral groups.
    Now, how these three aren't isomorphic between them? Easy: count number of elements. For example, \mathbb{Z}_8 has an element of order 8 and none of the other groups has such an element, and all the elements in the third group have order 2, but not all in the second one...

    Tonio
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 6
    Last Post: March 6th 2011, 11:08 AM
  2. Groups of Order 6
    Posted in the Advanced Algebra Forum
    Replies: 6
    Last Post: December 18th 2009, 01:17 PM
  3. Non-isomorphic groups of order 42.
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: February 15th 2009, 09:06 AM
  4. Order of groups involving conjugates and abelian groups
    Posted in the Advanced Algebra Forum
    Replies: 5
    Last Post: February 5th 2009, 09:55 PM
  5. Order in two groups
    Posted in the Statistics Forum
    Replies: 4
    Last Post: October 4th 2008, 05:12 PM

Search Tags


/mathhelpforum @mathhelpforum