Free equivariant extensors

We prove for a finite group G and a compact metric G -space Y that the conditions (1) Y∈LCn−1∩Cn−1 , and (2) Y∈G -AE(X) , for every normal n -dimensional space X endowed with a free numerable action of the group G , are equivalent. orollary we obtain: (A) For the space X endowed with a free action of the finite group G the conditions (1) the space X is normal, dim X⩽n and K(X;G)⩽n+1 ; (2) the space X is normal, dim X⩽n and K(X;G)<∞ ; (3) G∗⋯∗G∈G -AE(X) , are equivalent. (B) For a paracompact space X with a free action of the finite group G the inequality K(X;G)⩽dim X+1 holds.