Results 1 to 2 of 2

Math Help - Equivalence relation and order of each equivalence class

  1. #1
    Junior Member
    Joined
    Dec 2008
    From
    Missouri
    Posts
    55

    Equivalence relation and order of each equivalence class

    Problem to solve:
    Let X and Y be nonempty subsets of a group G, and XY denotes the set {g \in G| g = xy for some x \in X, y \in Y}.
    Let H and K be subgroups of a finite group G and define a relation on the set H \times K by setting (r,s) ~ (t,u) if rs = tu \forall r, t \in H, s,u \in K.
    Part 1: Show that this is an equivalence relation such that each equivalence class has order |H \cap K|.
    Part 2: Prove |HK| \cdot |H \cap K| = |H| \cdot |K|

    Hello,
    I need help with the problems above. I'm also a little confused on part 1...I know how to show the equivalence relation for the set H \times K, but then the problem stated something about the order of equivalence class. How am I going to show that? Appreciate anyone's help on this. Thank you in advance
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,326
    Thanks
    1298
    The "order of the equivalence classes" is just the number of elements in each equivalence class. And it is easy to prove that all equivalence classes, for a given equivalence relation, have the same number of elements so your answer will be a single number. Since you are dealing with groups, it might be simplest to find how many pairs (r, k) are equivalent to [itex](e_H, e_K)[/itex], the identity elements of each group.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. equivalence relation and equivalence classes
    Posted in the Discrete Math Forum
    Replies: 6
    Last Post: January 7th 2010, 06:36 PM
  2. Equivalence relation and Equivalence classes?
    Posted in the Discrete Math Forum
    Replies: 3
    Last Post: January 7th 2009, 03:39 AM
  3. class of equivalence
    Posted in the Number Theory Forum
    Replies: 2
    Last Post: January 4th 2009, 04:55 AM
  4. need help with equivalence class
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: December 18th 2008, 11:35 PM
  5. equivalence relation and class
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: September 20th 2008, 05:01 PM

Search Tags


/mathhelpforum @mathhelpforum