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An Operand for a Group of Order 1512
Author(s) -
Edge W. L.
Publication year - 1973
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/s2-7.1.101
Subject(s) - order (exchange) , operand , citation , group (periodic table) , enhanced data rates for gsm evolution , computer science , library science , information retrieval , mathematics , arithmetic , artificial intelligence , physics , quantum mechanics , finance , economics
1. The tableau on p. 105 built with the digits 0, 1,2, 3, 4, 5, 6, 7, 8 is one of a possible 1920 and was compiled in the year 1963 at about the time when [5] was published. Whether it is new or not, its construction, described below, with the help of the geometry of Study's quadric S, in projective space of 7 dimensions over the finite field GF(2), may be. One can observe the visual effects on the tableau of 1512 even permutations of the group H for which it is, as a whole, invariant. H could, in this context, be identified as the intersection of three alternating groups of degree 9 derivable from each other by using the triality on S. But H and its simple subgroup h of order 504, isomorphic to the linear fractional group LF(2, 2), occur also in a less elaborate geometry. LF(2, 2) is the group of projectivities on a 9-point line A and is extended to a group isomorphic to H by the automorphism of period 3 of GF(2) which, replacing each mark by its square, leaves three points (those with parameters 0,1, oo) of A unmoved while permuting the others in two cycles of three—a permutation unattainable by projectivities because any projectivity which leaves three points all unmoved can only be the identity.