Terminology: groups whose elements are their own actions

Question

Consider groups (or more generally monoids or semigroups) whose elements are functions and whose action $T$ is defined by the formula $T(s,x)=s(x)$.

Is there a name for such groups "whose elements are their own actions"?

Answer 1

For the case of a group action, let me suppose that the mathematical object on which the group is acting has been specified, i.e. the set of all elements $X = \{x\}$ equipped with whatever mathematical structure is appropriate, for example: a topology; or a vector space structure; or whatever. In that case I would refer to this kind of group action as a subgroup of $\textbf{Aut}(X)$, where $\text{Aut}(X)$ is the group of automorphisms of $X$, i.e. self-isomorphisms.

For the case of a monoid action, one uses $\text{End}(X)$ for the monoid of self-endomorphisms of $X$, and then your action could be called a submonoid of $\textbf{End}(X)$.

I'll say that the full information being used here to define $\text{Aut}(x)$ and $\text{End}(X)$ is a category of which $X$ is an object, for example: the category of topological spaces and continuous maps; or the category of vector spaces and linear maps over some field; or whatever.

Answer 2

Any group can be viewed in this way: identify an element $g$ with its corresponding left action $g(x)=g \cdot x$.