Abstract A generating set for a finite group G is minimal if no proper subset generates , and $m(G)$ denotes the maximal size of . We prove conjecture Lucchini, Moscatiello Spiga by showing that there exist $a,b> 0$ such any satisfies $m(G) \leqslant \cdot \delta (G)^b$ $\delta (G) = \sum _{p \text { prime}} m(G_p)$ where $G_p$ Sylow p -subgroup To do this, we first bound all almost simple g...

We find some new one-parameter families of exponential sums in every odd characteristic whose geometric and arithmetic monodromy groups are G2. 1. The exceptional identities Fix a prime p, and consider the p-Witt vectors of length 2 as a ring scheme over Z. The addtion law is given by (x, a) + (y, b) := (x+ y, a+ b+ (x + y − (x+ y))/p). The multiplication law is given by (x, a)(y, b) := (xy, xb...

Fix α ∈ ( 0 , 1 <mml:mrow class="MJX-TeXAtom-ORD"...

Let A be an isogeny class of abelian surfaces over Fq with Weil polynomial x4+ax3+bx2+aqx+q2. We show that A does not contain a surface that has a principal polarization if and only if a2 − b = q and b < 0 and all prime divisors of b are congruent to 1 modulo 3.

1 Definitions Definition 1. A graded ring is a ring S together with a set of subgroups Sd, d ≥ 0 such that S = ⊕ d≥0 Sd as an abelian group, and st ∈ Sd+e for all s ∈ Sd, t ∈ Se. One can prove that 1 ∈ S0 and if S is a domain then any unit of S also belongs to S0. A homogenous ideal of S is an ideal a with the property that for any f ∈ a we also have fd ∈ a for all d ≥ 0. A morphism of graded r...

Assume that $A$‎, ‎$B$ are Banach algebras and that $m:Atimes Brightarrow B$‎, ‎$m^prime:Atimes Arightarrow B$ are bounded bilinear mappings‎. ‎We study the relationships between Arens regularity of $m$‎, ‎$m^prime$ and the Banach algebras $A$‎, ‎$B$‎. ‎For a Banach $A‎$‎-bimodule $B$‎, ‎we show that $B$ factors with respect to $A$ if and only if $B^{**}$ is unital as an $A^{**}‎$‎-module‎. ‎Le...

A couple of years later, Carlitz [6] refined this theorem a little basically by dropping the conditions that f(0) = 0 and f(1) = 1 and by allowing f(a)−f(b) a−b to always be a square or always be a nonsquare, whenever a 6= b. He then accordingly modified the conclusion to f(x) = ax + b with a being a square or nonsquare according to which f(a)−f(b) a−b was. Now a change of notation. Let F = GF ...

Using elementary methods, the following results are obtalned:(1) If p is n n-m prime, 0 m n, 0 < b < ap n-m, and p ab, then (m) (-I)P-I(apb (rood pn). 2 A2_4B If r,s are the roots of x Ax-B, where (A,B) and D > 0, if n n u v rn+s n, and k > O, then (II) v =v (rood pn). n r-s n kpn kpn-1 (III) If p is odd and p D, then u (_D) u (rood pn); kp n P kp n-I (IV) u (_1)Bu n) n n-I (rood 2 k2 k2