Group of monomial functions over a group.

Question

Define a monomial function $f: G \to G$ to be a function that can be written in the form $f(x) = g x g'$ for some $g, g' \in G$.

Let $G$ be a finite group and consider the set of $S$ of all monomial functions from $G$ to $G$ that take one argument. For instance $(f(x) = abxcd) \in S$. Now if $f \in S$, then there exists $g \in S$ such that $f\circ g(x) =\text{id}(x)$. For the example given, $g(x) = (ab)^{-1} x (cd)^{-1}$. If we identify all monomial functions that always have the same value on $G$, then $S$ is a group under function composition.

What is this group called?

Answer 1

not sure of the name of your group, but is it related to a central product of two copies of $G$?

if $|G|=n$ then there is an injective homomorphism $\rho:G \to S_n(G)$ with $\rho(g)(x)=gx$

similarly there is an injective homomorphism $\lambda:G \to S_n(G)$ with $\lambda(g')(x) = xg'$

the groups $\lambda(G),\rho(G)$ commute in $S_n$ and they intersect in a subgroup of $S_n(G)$ isomorphic to $Z(G)$.