Design isomorphisms and group isomorphisms

In this paper conditions are given which imply that a design isomorphism between designs with group actions is in fact a group isomorphism. The conditions are geometric. Automorphism groups are then calculated for a family of designs using the geometric conditions.The research in the paper was conducted at Loyola University of Chicago.     

The minimum principle for affine functions and isomorphisms of continuous affine function spaces

Let X be a compact convex set and let ext X stand for the set of extreme points of X. We show that if $$f:Xrightarrow {mathbb {R}}$$ is an affine function with the point of continuity property such that $$fle 0$$ on $${text {ext}},X$$, then $$fle 0$$ on X. As a corollary of this minimum principle, we obtain a generalization of a theorem by C.H. Chu and H.B. Cohen by proving the following result. Let X, Y be compact convex sets such that every extreme point of X and Y is a weak peak point and let $$T:mathfrak {A}^c(X)rightarrow mathfrak {A}^c(Y)$$ be an isomorphism such that $$left| Tright| cdot left| T^{-1}right| <2$$. Then $${text {ext}},X$$ is homeomorphic to $${text {ext}},Y$$.     

Semigroups of the basic affine lie group and their automorphisms

Omsk. Translated fromSibirskiî Matematicheskiî Zhurnal, Vol. 33, No. 4, pp. 59–64, July–August, 1992.     

Remarks on affine complete distributive lattices

We characterise the Priestley spaces corresponding to affine complete bounded distributive lattices. Moreover we prove that the class of affine complete bounded distributive lattices is closed under products and free products. We show that every (not necessarily bounded) distributive lattice can be embedded in an affine complete one and that ℚ ∩ [0, 1] is initial in the class of affine complete lattices.     

Roots of the affine Cremona group

Let k ^[n] = k[x ₁,…, x _n ] be the polynomial algebra in n variables and let \mathbbAⁿ = \textSpec  \boldk^{[ n ]} {\mathbb{A}^n} = {\text{Spec}}\;{{\bold{k}}^{\left[ n \right]}} . In this note we show that the root vectors of \textAu\textt^*( \mathbbAⁿ ) {\text{Au}}{{\text{t}}^*}\left( {{\mathbb{A}^n}} \right) , the subgroup of volume preserving automorphisms in the affine Cremona group \textAut( \mathbbAⁿ ) {\text{Aut}}\left( {{\mathbb{A}^n}} \right) , with respect to the diagonal torus are exactly the locally nilpotent derivations x ^α (∂/∂x _i ), where x ^α is any monomial not depending on x _i . This answers a question posed by Popov.     

Mendelsohn designs admitting the affine group

All Mendelsohn designs containing a Frobenius group with cyclic complement of orderv – 1 as a subgroup of the automorphism are found. Furthermore, the automorphism group of each of the designs is constructed. These designs generalize Mendelsohn's construction of Mendelsohn designs containing a certain doubly transitive automorphism group.The research on this paper was partially supported by North Texas State Faculty Research Grant #35524.     

Alternating group covers of the affine line

For an odd prime p t= 2 mod 3, we prove Abhyankar’s Inertia Conjecture for the alternating group A _p+2, by showing that every possible inertia group occurs for a (wildly ramified) A _p+2-Galois cover of the projective k-line branched only at infinity where k is an algebraically closed field of characteristic p > 0. More generally, when 2 ≤ s < p and gcd(p−1, s+1) = 1, we prove that all but finitely many rational numbers which satisfy the obvious necessary conditions occur as the upper jump in the filtration of higher ramification groups of an A _p+s -Galois cover of the projective line branched only at infinity.