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Thread: Stabilizers

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    Forum Admin topsquark's Avatar
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    Stabilizers

    Okay, just to make sure I have this straight:

    Given a group G and an element g of G

    The "stabilizer" of g is the subset $\displaystyle \{x \} \subset G$ such that $\displaystyle yg = g$ for all $\displaystyle y \in \{ x \}$.

    And the stabilizer of g is a subgroup of G, right?

    Thanks!
    -Dan
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    Quote Originally Posted by topsquark View Post
    Okay, just to make sure I have this straight:

    Given a group G and an element g of G

    The "stabilizer" of g is the subset $\displaystyle \{x \} \subset G$ such that $\displaystyle yg = g$ for all $\displaystyle y \in \{ x \}$.

    And the stabilizer of g is a subgroup of G, right?

    Thanks!
    -Dan
    You got the elements of the group confused with elements of the set.

    Let $\displaystyle G$ be a a group, and let $\displaystyle X$ be a set. Let $\displaystyle G$ act on $\displaystyle X$.
    Given an element $\displaystyle x\in X$ the stabalizer, $\displaystyle G_x$, is the set $\displaystyle \{g\in G|gx=x\}$. And yes, the stabalizer is a subgroup.

    I can give some examples if you want to.
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    Forum Admin topsquark's Avatar
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    Quote Originally Posted by ThePerfectHacker View Post
    You got the elements of the group confused with elements of the set.

    Let $\displaystyle G$ be a a group, and let $\displaystyle X$ be a set. Let $\displaystyle G$ act on $\displaystyle X$.
    Given an element $\displaystyle x\in X$ the stabalizer, $\displaystyle G_x$, is the set $\displaystyle \{g\in G|gx=x\}$. And yes, the stabalizer is a subgroup.

    I can give some examples if you want to.
    Ahhh! The lightbulb just went on. I think I've got it now. I'll play around with it a bit and let you know if I need more. Thanks!

    -Dan
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