Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userNumbers which are only orders of abelian or nilpotent groups
84
0
0.0
(
0
)
Authors
Matthew Just
Ask ChatGPT about the research
No Arabic abstract
Refining a result of Erdos and Mays, we give asymptotic series expansions for the functions $A(x)-C(x)$, the count of $nleq x$ for which every group of order $n$ is abelian (but not all cyclic), and $N(x)-A(x)$, the count of $nleq x$ for which every group of order $n$ is nilpotent (but not all abelian).
rate research
Read More
We call $n$ a cyclic number if every group of order $n$ is cyclic. It is implicit in work of Dickson, and explicit in work of Szele, that $n$ is cyclic precisely when $gcd(n,phi(n))=1$. With $C(x)$ denoting the count of cyclic $nle x$, ErdH{o}s proved that $$C(x) sim e^{-gamma} x/logloglog{x}, quadtext{as $xtoinfty$}.$$ We show that $C(x)$ has an asymptotic series expansion, in the sense of Poincare, in descending powers of $logloglog{x}$, namely $$frac{e^{-gamma} x}{logloglog{x}} left(1-frac{gamma}{logloglog{x}} + frac{gamma^2 + frac{1}{12}pi^2}{(logloglog{x})^2} - frac{gamma^3 +frac{1}{4} gamma pi^2 + frac{2}{3}zeta(3)}{(logloglog{x})^3} + dots right). $$
The article deals with profinite groups in which the centralizers are abelian (CA-groups), that is, with profinite commutativity-transitive groups. It is shown that such groups are virtually pronilpotent. More precisely, let G be a profinite CA-group. It is shown that G has a normal open subgroup N which is either abelian or pro-p. Further, a rather detailed information about the finite quotient G/N is obtained.
In this paper we study virtual rational Betti numbers of a nilpotent-by-abelian group $G$, where the abelianization $N/N$ of its nilpotent part $N$ satisfies certain tameness property. More precisely, we prove that if $N/N$ is $2(c(n-1)-1)$-tame as a $G/N$-module, $c$ the nilpotency class of $N$, then $mathrm{vb}_j(G):=sup_{Minmathcal{A}_G}dim_mathbb{Q} H_j(M,mathbb{Q})$ is finite for all $0leq jleq n$, where $mathcal{A}_G$ is the set of all finite index subgroups of $G$.
Motivated by the construction of Bach flat neutral signature Riemannian extensions, we study the space of parallel trace free tensors of type $(1,1)$ on an affine surface. It is shown that the existence of such a parallel tensor field is characterized by the recurrence of the symmetric part of the Ricci tensor.
We classify all triples $(G,V,H)$ such that $SL_n(q)leq Gleq GL_n(q)$, $V$ is a representation of $G$ of dimension greater than one over an algebraically closed field $FF$ of characteristic coprime to $q$, and $H$ is a proper subgroup of $G$ such that the restriction $Vdar_{H}$ is irreducible. This problem is a natural part of the Aschbacher-Scott program on maximal subgroups of finite classical groups.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
