Let $E$ be a set equipped with a binary operation $*$ which is associative.
Assume that for any $a$ and $y$ in $E$, there exists $x$ in $E$ such that
$$y=a*x*a$$
Prove that $(E,*)$ has an identity element.
I don't know where / how to start, thanks for any hints.
My first move is always to try something simple, with fewer moving parts. So let me first see what happens if $a = y$. Then the condition becomes: For any $a$, there is an $x$ such that $a = a*x*a$. In that case, if there is to be an identity, $a*x$ would be a good candidate. Can I perhaps show that $a*x*b = b$? Not without some way of relating $a$ and $b$. But I can use the condition again: There is a $y$ such that $b = a*y*a$. Let's try that: $$ \begin{align} a*x*b &= (a*x)*b\\ &= (a*x)*(a*y*a)\\ &= (a*x*a)*y*a\\ &= a*y*a\\ &= b. \end{align} $$ How about that! $a*x$ is, at least, a left identity.
You can take it from there.