Results 1 to 2 of 2

Math Help - Isomorphism

  1. #1
    Junior Member
    Joined
    Aug 2008
    Posts
    44

    Isomorphism

    Find the minimum of all positive integers m such that every sigma in S_9 (Sigma in A_9) satisfies sigma^ n = 1.

    Find 4 different subgroups of S_4, isomorphic to S_3, and nine isomorphic to S_2. ( I dont need all the subgroups just how to find them isomorphic)

    Let G be a subgroup of S_5. Prove that if G contains a 5-Cycle and a 2-Cycle, then G = S_5.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Swlabr's Avatar
    Joined
    May 2009
    Posts
    1,176
    Quote Originally Posted by Juancd08 View Post
    Find the minimum of all positive integers m such that every \sigma \in S_9 ( \sigma \in A_9) satisfies \sigma^ n = 1.
    Hint: for \sigma = (a_1 a_2 \ldots a_i) a permutation of i objects then \sigma has order precisely i. Prove this, then apply it to your questions.

    Quote Originally Posted by Juancd08 View Post
    Find 4 different subgroups of S_4, isomorphic to S_3, and nine isomorphic to S_2.
    To find the subgroups isomorphic to S_2 note that S_2 \equiv C_2 (=\mathbb{Z}_2), the cyclic group of 2 elements. So, how many elements in this group have order 2?...

    For S_3, your groups are just the permutations of 3 elements, \{a_1, a_2, a_3\} \subset \{1,2,3,4\}. This is because S_3 is merely the group of all possible permutations of 3 objects, by it's "rawest" definition. If you take all possible permutations of 3 objects you will get a group isomorphic to S_3.

    Quote Originally Posted by Juancd08 View Post
    Let G be a subgroup of S_5. Prove that if G contains a 5-Cycle and a 2-Cycle, then G = S_5.
    I suspect someone here can provide you with a hint towards a much neater proof than I have. Sorry. However, I would advise you to try this for smaller values of n to see how it works. Note that S_n is generated by a transposition and an n-cycle for all n > 2
    Last edited by Swlabr; June 28th 2009 at 11:27 PM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Isomorphism
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: December 12th 2011, 01:57 AM
  2. Isomorphism
    Posted in the Advanced Algebra Forum
    Replies: 10
    Last Post: October 27th 2010, 12:08 AM
  3. isomorphism
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: September 30th 2010, 09:52 AM
  4. isomorphism
    Posted in the Number Theory Forum
    Replies: 2
    Last Post: March 10th 2010, 08:50 AM
  5. Replies: 4
    Last Post: February 14th 2010, 03:05 AM

Search Tags


/mathhelpforum @mathhelpforum