Knowledge Bank

A generalization of the standard semi-direct product of groups is given. The following special case is exploited in the construction of partial 4-gons. Let G be the set of 4-tuples of elements of the finite field F. For all i, j with l < i , j<2, let Ljj and Rij be linear transformations of F over its prime subfield. Then define a product on G as follows: (a1, b1, c1, d1)- (a2, b2, c2, d2) = (ai+a2, b1+b2, L11 R11 L12 R12 L21 R2i L22 R22 a1 b2 +a2 b1 +c1+c2, a1 b2 +a2 b1 +d1+d2). With this product G is a group. Let A and B be the subgroups of G consisting of elements of the form (a, 0, 0, 0), a e F, and (0, b, 0,0), b e F, respectively. Then necessary and sufficient conditions on Lij and Rij are found for the coset geometry ir(G, A, B) to be a partial generalized 4-gon.