By René Schoof
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Example text
Special attention is given to the existence of a ‘Foulser cover’ for rational ‘Desarguesian’ nets. Roughly speaking, a Foulser cover is a partition of the points covered by a partial spread consisting of subspaces that are subplanes. There is a comprehensive and complete theory of Foulser-covers for completely arbitrary nets, based on the classification of derivation arising from Prohaska and Cofman, the more recent work of Johnson, Thas and De Clerck, and culminating in the beautiful and surprising theorem of Johnson [753].
Thus, the slope sets of spreads yield spread sets, and conversely every spread set may be regarded as a slope set of a given spread. So we may regard slope sets and spread sets as being conceptually synonymous: slope sets correspond to a generic construction for sets of linear maps that satisfy the axioms for spread sets. We may get new spread sets, from the given spread set σ, in several ways. For example, if we simply choose a pair of GF (p)-linear bijections, A : S → X and B : Y → T , then AσB is a spread set: we shall consider spread sets as being equivalent—by basis-change—if they are related in this way.
Now there are two projective spaces associated with V and/or AG(V, K): (i) Extend AG(V, K) to a projective space by the method of adjunction of a ‘hyperplane at infinity’. We shall call this projective space P G(V, K). (ii) Form the projective space obtained from V by taking the ‘points’ to be the 1-dimensional K-subspaces and the set of ‘projective subspaces’ to be the lattice 26 4. PARTIAL SPREADS AND GENERALIZATIONS. of vector subspaces. Let V = W ⊕ K, and denote W by V − and V by W + . We shall use the notation P G(V − , K) to denote this projective space.
Algebra 2. The symmetric groups Sn by René Schoof
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