The Existence of T
نویسنده
چکیده
Let ~ be a (possibly infinite) projective plane, and G a collineation group of ~ . Suppose temporarily that ~ is a translation plane with respect to L, and G is the group of collineations fixing L. Then G=TGx, where T is the translation group with respect to L, and x is an affine point. If, moreover, the characteristic of ~ is not 2, then Gx= CG(t) and G= TCa(t), where t is an involutory homology with center x. In this note we will consider a sort of converse of these facts. Take afinite collineation group G generated by involutory homologies, where ~ is again any projective plane. We will assume that G has an involutory homology t whose behavior resembles that of the t in the preceding paragraph, and then deduce the existence of a factorization G = TCa (t) as above. The precise statement is as follows. Let 0(G) denote the largest normal subgroup of G of odd order. Define Z*(G)>_O(G) by: Z*(G)/O(G) is the center of G/O(G).
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