We give the very first example of a simple 2−(81,6,2) design. Its points are elements elementary abelian group order 81 and each block is union two parallel lines 4-dimensional geometry over field 3. Hence it also additive.

We give an explicit description of the tracial state simplex C ⁎ -algebra ( G ) arbitrary connected, second countable, locally compact, solvable group . show that every lifts from a abelianized group, and intersection kernels all states is proper ideal unless abelian. As consequence, connected nonabelian Lie cannot embed into simple unital AF-algebra.

Abstract In this paper we study the relativized Lascar Galois group of a strong type. The is quasi-compact connected topological group, and if in addition underlying theory T G -compact, then compact. We apply compact to obtain model theoretic results note. For example, use divisibility type show that, simple theory, such types have certain property that call divisible amalgamation. main result...

Let P2(G) be defined as the probability that any two elements selected at random from the group G, commute with one another. If G is an Abelian group, P2(G) = 1, so our interest lies in the properties of the commutativity of nonAbelian groups. Particular results include that the maximum commutativity of a non-Abelian group is 5/8, and this degree of commutativity only occurs when the order of t...

Let $G$ be a finite group. We construct the prime graph of $ G $,which is denoted by $ Gamma(G) $ as follows: the vertex set of thisgraph is the prime divisors of $ |G| $ and two distinct vertices $ p$ and $ q $ are joined by an edge if and only if $ G $ contains anelement of order $ pq $.In this paper, we determine finite groups $ G $ with $ Gamma(G) =Gamma(L_3(q)) $, $2 leq q < 100 $ and prov...