Set-Theoretic Solutions of the Pentagon Equation

set-theoretic solution of the Pentagon Equation on a non-empty set S is a map s:S^2→ S^2 such that s_23s_13s_12=s_12s_23 , where s_12=s× id , s_23= id × s and s_13=(τ× id )( id × s)(τ× id ) are mappings from S^3 to itself and τ :S^2→ S^2 is the flip map, i.e., τ (x,y) =(y,x) . We give a description of all involutive solutions, i.e., s^2= id . It is shown that such solutions are determined by a factorization of S as direct product X× A × G and a map σ :A→ 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 σ is irrelevant. In particular, if S is finite of cardinality 2^n(2m+1) for some n,m⩾ 0 then, on S , there are precisely ( [ n+2; 2 ]) non-isomorphic solutions of the Pentagon Equation.