Block Transitive Resolutions of t-designs and Room Rectangles

By a resolution of t-designs we mean a partition of the trivial design \Gamma X k \Delta of all k-subsets of a v-set X into t \Gamma (v 0 ; k; ) designs, where v 0 v. A resolution of t-designs with v = v 0 is also called a large set of t-designs. A Room rectangle R, based on \Gamma X k \Delta , is a rectangular array whose non-empty entries are k-sets. This array has the further property that taken together the rows form a resolution of t 1 \Gammadesigns, and the columns form a resolution of t 2 \Gammadesigns. A resolution of t\Gammadesigns for \Gamma X k \Delta is said to admit G as a block transitive automorphism group if G is k\Gammahomogeneous on X, and permutes the t\Gammadesigns of the resolution among themselves. Some examples of block transitive resolutions of nontrivial t-designs, for t 2, are: 1) an M 11 -invariant set of 3-(10, 4, 1) designs, 2) an M 12 - invariant set of 4-(11, 5, 1) designs, 3) an M 24 -invariant set of 2-(21, 5, 1) designs, 4) a P \Gam..

The tits alternative for groups defined by periodic paired relations

The class of groups defined by periodic paired relations, as introduced by Vinberg, includes the generalized triangle groups, the generalized tetrahedron groups, and the generalized Coxeter groups. We observe that any group defined by periodic paired relations Gamma can be realized as a so-called '' Pride group ''. Using results of Howie and Kopteva we give necessary and sufficient conditions for this Pride group to be non-spherical. Under such conditions, we show that Gamma satisfies the Tits alternative

Legislative Documents

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