Let
$p$
be an odd prime. We construct a
$p$
-group
$P$
of nilpotency class two, rank seven and exponent
$p$
, such that
$\text{Aut}(P)$
induces
...$N_{\text{GL}(7,p)}(G_{2}(p))=Z(\text{GL}(7,p))G_{2}(p)$
on the Frattini quotient
$P/\unicodeSTIX{x1D6F7}(P)$
. The constructed group
$P$
is the smallest
$p$
-group with these properties, having order
$p^{14}$
, and when
$p=3$
our construction gives two nonisomorphic
$p$
-groups. To show that
$P$
satisfies the specified properties, we study the action of
$G_{2}(q)$
on the octonion algebra over
$\mathbb{F}_{q}$
, for each power
$q$
of
$p$
, and explore the reducibility of the exterior square of each irreducible seven-dimensional
$\mathbb{F}_{q}G_{2}(q)$
-module.
We prove transitivity for volume-preserving
$C^{1+}$
diffeomorphisms on
$\mathbb{T}^{3}$
which are isotopic to a linear Anosov automorphism along a path of weakly partially hyperbolic diffeomorphisms.
Let ð´ be a finite group acting by automorphisms on the finite group ðº. We introduce the commuting graph Image omitted of this action and study some questions related to the structure of ðº ...under certain graph theoretical conditions on Image omitted.
In this paper, we prove the equidistribution of saddle periodic points for Hénon-type automorphisms of C k with respect to its equilibrium measure. A general strategy to obtain equidistribution ...properties in any dimension is presented. It is based on our recent theory of densities for positive closed currents. Several fine properties of dynamical currents are also proved.
We develop an approach to the longstanding conjecture of Kierstead concerning the character of strongly nontrivial automorphisms of computable linear orderings. Our main result is that for any η-like ...computable linear ordering B, such that B has no interval of order type η, and such that the order type of B is determined by a -limitwise monotonic maximal block function, there exists computable L B such that L has no nontrivial Π 1 0 automorphism.
Abstract
In this paper we prove that surfaces of general type with irregularity
$q\geq 3$
are rationally cohomologically rigidified, and so are minimal surfaces
$S$
with
$q(S)= 2$
unless
${ ...K}_{S}^{2} = 8\chi ({ \mathcal{O} }_{S} )$
. Here a surface
$S$
is said to be rationally cohomologically rigidified if its automorphism group
$\mathrm{Aut} (S)$
acts faithfully on the cohomology ring
${H}^{\ast } (S, \mathbb{Q} )$
. As examples we give a complete classification of surfaces isogenous to a product with
$q(S)= 2$
that are not rationally cohomologically rigidified.
In this paper, we study the supercharacter theories of elementary abelian \(p\)-groups of order \(p^{2}\). We show that the supercharacter theories that arise from the direct product construction and ...the \(\ast\)-product construction can be obtained from automorphisms. We also prove that any supercharacter theory of an elementary abelian \(p\)-group of order \(p^{2}\) that has a non-identity superclass of size \(1\) or a non-principal linear supercharacter must come from either a \(\ast\)-product or a direct product. Although we are unable to prove results for general primes, we do compute all of the supercharacter theories when \(p = 2,\, 3,\, 5\), and based on these computations along with particular computations for larger primes, we make several conjectures for a general prime \(p\).
For each prime
$p$
we construct a family
$\{G_{i}\}$
of finite
$p$
-groups such that
$|\text{Aut}(G_{i})|/|G_{i}|$
tends to zero as
$i$
tends to infinity. This disproves a well-known conjecture that
...$|G|$
divides
$|\text{Aut}(G)|$
for every nonabelian finite
$p$
-group
$G$
.
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