اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدIdentifying measures on non-abelian groups and modules by their moments via reduction to a local problem
186
0
0.0
(
0
)
تأليف
Will Sawin
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
Work on generalizations of the Cohen-Lenstra and Cohen-Martinet heuristics has drawn attention to probability measures on the space of isomorphism classes of profinite groups. As is common in probability theory, it would be desirable to know that these measures are determined by their moments, which in this context are the expected number of surjections to a fixed finite group. We show a wide class of measures, including those appearing in a recent paper of Liu, Wood, and Zurieck-Brown, have this property. The method is to work locally with groups that are extensions of a fixed group by a product of finite simple groups. This eventually reduces the problem to the case of powers of a fixed finite simple group, which can be handled by a simple explicit calculation. We can also prove a similar theorem for random modules over an algebra.
قيم البحث
اقرأ أيضاً
We present a generic algorithm for computing discrete logarithms in a finite abelian p-group H, improving the Pohlig-Hellman algorithm and its generalization to noncyclic groups by Teske. We then give a direct method to compute a basis for H without
using a relation matrix. The problem of computing a basis for some or all of the Sylow p-subgroups of an arbitrary finite abelian group G is addressed, yielding a Monte Carlo algorithm to compute the structure of G using O(|G|^0.5) group operations. These results also improve generic algorithms for extracting pth roots in G.
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).
A $Gamma$-magic rectangle set $MRS_{Gamma}(a, b; c)$ of order $abc$ is a collection of $c$ arrays $(atimes b)$ whose entries are elements of group $Gamma$, each appearing once, with all row sums in every rectangle equal to a constant $omegain Gamma$
and all column sums in every rectangle equal to a constant $delta in Gamma$. In this paper we prove that for ${a,b} eq{2^{alpha},2k+1}$ where $alpha$ and $k$ are some natural numbers, a $Gamma$-magic rectangle set MRS$_{Gamma}(a, b;c)$ exists if and only if $a$ and $b$ are both even or and $|Gamma|$ is odd or $Gamma$ has more than one involution. Moreover we obtain sufficient and necessary conditions for existence a $Gamma$-magic rectangle MRS$_{Gamma}(a, b)$=MRS$_{Gamma}(a, b;1)$.
We deal with some pcf investigations mostly motivated by abelian group theory problems and deal their applications to test problems (we expect reasonably wide applications). We prove almost always the existence of aleph_omega-free abelian groups with
trivial dual, i.e. no non-trivial homomorphisms to the integers. This relies on investigation of pcf; more specifically, for this we prove that almost always there are F subseteq lambda^kappa which are quite free and has black boxes. The almost always means that there are strong restrictions on cardinal arithmetic if the universe fails this, the restrictions are everywhere. Also we replace Abelian groups by R-modules, so in some sense our advantage over earlier results becomes clearer.
In previous work, the authors confirmed the speculation of J. G. Thompson that certain multiquadratic fields are generated by specified character values of sufficiently large alternating groups $A_n$. Here we address the natural generalization of thi
s speculation to the finite general linear groups $mathrm{GL}_mleft(mathbb{F}_qright)$ and $mathrm{SL}_2left(mathbb{F}_qright)$.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
