A contraction of square transversal designs

In this paper we show that if a square transversal design TDλ[k;u]TDλ[k;u], say D(=(P,B))D(=(P,B)), admits a class semiregular automorphism group G of order s  , then we have a |P|s by |P|s matrix M   with entries from G∪{0}G∪{0} satisfying (*)M(M(-1))T=(Aij)1⩽i,j⩽m, where Aij=λuIt(t=us), if i=ji=j, and λG^Jt,t, otherwise. As an application of (*)(*), we show that any symmetric TD2[12;6]TD2[12;6] admits no nontrivial elation. We also obtain a result that gives us a restriction on the existence of elations of putative projective planes of composite order.