A set-theoretic solution of the Pentagon Equation on a non-empty set $S$ is a map $s\colon S^2\to S^2$ such that $s_{ 23}s_{ 13}s_{ 12}=s_{ 12}s_{ 23}$, where $s_{ 12}=s\times\mathrm{ id}$, $s_{ 23}=\mathrm{ id}\times s$ and $s_{ 13}=(\tau\times\mathrm{ id})(\mathrm{ id}\times s)(\tau\times\mathrm{ id})$ are mappings from $S^3$ to itself and $\tau\colon S^2\to S^2$ is the flip map, i.e., $\tau (x,y) =(y,x)$. We give a description of all involutive solutions, i.e., $s^2=\mathrm{ id}$. It is shown that such solutions are determined by a factorization of $S$ as direct product $X\times A \times G$ and a map $\sigma\colon A\to\mathrm{ Sym}(X)$, where $X$ is a non-empty set and $A,G$ are elementary abelian $2$-groups. Isomorphic solutions are determined by the cardinalities of $A$, $G$ and $X$, i.e., the map $\sigma$ is irrelevant. In particular, if $S$ is finite of cardinality $2^n(2m+1)$ for some $n,m\geq 0$ then, on $S$, there are precisely $\binom{ n+2}{ 2}$ non-isomorphic solutions of the Pentagon Equation.
Categories: math.RA