Prove that $[K : aHa^{-1} \cap K]$ is equal to number of left cosets of $H$ in $KaH$, where $K, H$ are subgroups of $G$ and $a\in G$.
My work:
If we use $g \in G$, such that $gH$ is left coset, $gH$ will be an element of $KaH$ if $g \in Ka$. From the second theorem of isomorphism, we have the following: $K / K \cap aHa^{-1} \cong KaHa^{-1} / aHa^{-1}$. If we prove that $gH \cong KaHa^{-1} / aHa^{-1}$ we will have what we need.
Let $l \in KaHa^{-1}$. That means $l$ can be presented as $l = k \cdot h$ where $k \in K$, $h \in aHa^{-1}$. Then, $laHa^{-1} \in KaHa^{-1} / aHa^{-1}$ is equivalent to $kaHa^{-1}$. So, every element in $ KaHa^{-1} / aHa^{-1}$ has form of $kaHa^{-1}$. Now we have to prove that that is isomorphic to $gH$.
Let $\phi : KaH \to KaHa^{-1}$ be defined as $\phi(kah) = kaha^{-1}$, where $a$ is fixed element. $\phi$ is an isomorphism, which implies requested.
Can someone please tell me if this proof is correct? If not, what is incorrect?
The elements $k_1,\ldots,k_t\in K$ are coset representatives for $aHa^{-1}\cap K = H_a$ in $K$ if and only if the elements $k_1a,\ldots,k_ta$ are representatives for the left $H$-cosets in $KaH$.
Proof. First notice that $x,y\in K$ are in the same $H_a$ coset if and only if $x^{-1}y = aha^{-1} \in H_a$ if and only if $ya = xah$ for $h\in H$, if and only if the left cosets $xaH$ and $yaH$ coincide.
This proves that the index $[K:H_a]$ is equal to the number of left $H$-cosets appearing in $KaH$, which is what you wanted.