Results 1 to 3 of 3

Math Help - Lagrange's Theorem

  1. #1
    Junior Member
    Joined
    Sep 2009
    Posts
    34

    Question Lagrange's Theorem

    Let G be a group and H a subgroup of G. Define, for a,b elements of G, a is an equivalence relation to b if a^-1b element of H. Prove that this defines an equivalence relation on G and show that [a] = aH = {ah,h is an element of H}. The sets aH are called left cosets of H in G
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Newbie lepton's Avatar
    Joined
    Sep 2009
    Posts
    10
    This is pretty straight forward, just check the three properties of an equivalence relation. Recall an equivalence relation is reflexive, symmetric, and transitive. Here's a start.

    Let a,b,c\in H

    If ab^{-1}\in H, then since H is a subgroup, (ab^{-1})^{-1}=ba^{-1}\in H (symmetry)

    Now you need only show:

    aa^{-1}\in H (reflexivity)

    and

    If ab^{-1}\in H and bc^{-1}\in H, then ac^{-1}\in H (transitivity)

    For the second part, show containment both ways, [a]\subset aH and aH\subset [a]. It can be done in one fell swoop with a short string of equivalences if implications in both directions seems redundant.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Sep 2009
    Posts
    34
    Thanks Lepton..
    I am stil having trouble showing that [a] is a subset of aH and aH is a subset of [a]..Please can you take me through the steps??
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. using the Lagrange Theorem
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: July 29th 2010, 02:09 PM
  2. Prove Wilson's theorem by Lagrange's theorem
    Posted in the Number Theory Forum
    Replies: 2
    Last Post: April 10th 2010, 02:07 PM
  3. Lagrange's Theorem(?)
    Posted in the Pre-Calculus Forum
    Replies: 1
    Last Post: January 14th 2010, 01:35 AM
  4. Lagrange's Mean Value Theorem
    Posted in the Calculus Forum
    Replies: 3
    Last Post: December 20th 2008, 07:31 AM
  5. Lagrange Theorem.
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: October 16th 2008, 08:04 PM

Search Tags


/mathhelpforum @mathhelpforum