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Thread: subgroup

  1. #1
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    subgroup

    My niece told me that she needed some help... so, here I am begging for yours!

    It's in spanish, so I'll translate as best as possible:

    Let H be a subgroup of a certain group G and $\displaystyle a$ a fixed element of G. Prove that $\displaystyle K$ - the set of all the elements in the form $\displaystyle aha^{-1}$ with $\displaystyle h \in H$ is a subgroup of G.

    $\displaystyle K$ ={$\displaystyle x \in G : x = aha^{-1}$, for some $\displaystyle h \in H$}
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  2. #2
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    Quote Originally Posted by Kimberly View Post
    My niece told me that she needed some help... so, here I am begging for yours!

    It's in spanish, so I'll translate as best as possible:

    Let H be a subgroup of a certain group G and $\displaystyle a$ a fixed element of G. Prove that $\displaystyle K$ - the set of all the elements in the form $\displaystyle aha^{-1}$ with $\displaystyle h \in H$ is a subgroup of G.

    $\displaystyle K$ ={$\displaystyle x \in G : x = aha^{-1}$, for some $\displaystyle h \in H$}
    $\displaystyle \forall x,y \in K, x = ah_1a^{-1} , y^{-1} = ah_{2}^{-1}a^{-1}$ for some $\displaystyle h_1, h_2 \in H$. $\displaystyle xy^{-1} = (ah_1a^{-1})(ah_{2}^{-1}a^{-1}) = ah_1h_{2}^{-1}a^{-1}$
    But since H is a subgroup, $\displaystyle h_1h_{2}^{-1} \in H$
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