Coset Geometries of Some Generalized Semidirect Products of Groups
نویسنده
چکیده
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: (ai, bi, ci, di)(a2, b2, c2, d2) = (ai+a2, bi+b2 l Ln Rn Li2 Ri2 L21 R2i L22 R22 ai b2 +a2 bi +ci+c2 , &i b2 +a2 bi +di+d 2 ) . 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 R;j are found for the coset geometry ir(G, A, B) to be a partial generalized 4-gon.
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تاریخ انتشار 2017