1. 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 $\displaystyle h \in H$}

2. Originally Posted by Kimberly
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, $\displaystyle h_1h_{2}^{-1} \in H$