# Math Help - subgroup

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

$K$ ={ $x \in G : x = aha^{-1}$, for some $h \in H$}

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

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