Results 1 to 3 of 3

Math Help - kenrnal of a map for group action

  1. #1
    Forum Admin topsquark's Avatar
    Joined
    Jan 2006
    From
    Wellsville, NY
    Posts
    9,846
    Thanks
    320
    Awards
    1

    kenrnal of a map for group action

    Okay, given a function for a group action of the group G acting on a set A (with a fixed point a of A): \phi _a : G \times \{ a \} \to A. One definition I have for the kernal is ker \phi _a = \{ (g, a) | \phi _a (g, a) = 1_A ~ \forall g \in G \}. But the definition in the group action section is that ker \phi _a = \{ g \in G | g \cdot a = a \}. Are these somehow equivalent? I don't see it.

    -Dan
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Member Haven's Avatar
    Joined
    Jul 2009
    Posts
    197
    Thanks
    8

    Re: kenrnal of a map for group action

    I don't understand the definition of \phi_a . If a is a fixed point of G, then g.a=a for all g. So what is phi_a(g,a) for g\in G
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Forum Admin topsquark's Avatar
    Joined
    Jan 2006
    From
    Wellsville, NY
    Posts
    9,846
    Thanks
    320
    Awards
    1

    Re: kenrnal of a map for group action

    Quote Originally Posted by Haven View Post
    I don't understand the definition of \phi_a . If a is a fixed point of G, then g.a=a for all g. So what is phi_a(g,a) for g\in G
    In general the group action of a group G acting on an element of a set A is defined as \phi : G \times A \to A :g \cdot a \mapsto a'. The group G essentially acts as a permutation on the set A. If we restrict ourselves to a fixed element a of A (not G) then we have \phi _a : G \times \{ a \} \to A. My question is how to define the kernal of \phi _a.

    -Dan
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Example of a group action
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: November 1st 2013, 09:22 PM
  2. Group action G on itself
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: October 21st 2011, 03:40 PM
  3. group action
    Posted in the Advanced Algebra Forum
    Replies: 5
    Last Post: August 26th 2010, 12:07 PM
  4. Group action
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: April 15th 2010, 12:49 PM
  5. Group action
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: October 15th 2009, 12:58 PM

Search Tags


/mathhelpforum @mathhelpforum