Capable two-generator 2-groups of class two
英文摘要:
A group is called capable if it is a central factor group. We characterize the capable 2-generator 2-groups of class 2 in terms of a standard presentation.
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Arturo Magidin
A group is called capable if it is a central factor group. We characterize the capable 2-generator 2-groups of class 2 in terms of a standard presentation.
机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。
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A group is called capable if it is a central factor group. For each prime $p$ and positive integer $c$, we prove the existence of a capable $p$-group of class $c$ minimally generated by an element of order $p$ and an element of order $p^{1+\lfloor\frac{c-1}{p-1}\rfloor}$. This is best possible.
Let p be a prime number. We give the explicit structure of 2- nilpotent multiplier for each finite 2-generator p-group of class two. Moreover, 2-capable groups in that class are characterized.
This note collects several results on the capability of $p$-groups of class two and prime exponent. Among the new results, we settle the 4-generator case for this class.
We consider the capability of $p$ groups of class two and odd prime exponent. We use linear algebra and counting arguments to establish a number of new results. In particular, we settle the 4-generator case, and prove a sufficient condition based on the ranks of $G/Z(G)$ and $[G,G]$.
We show that if $G$ is any $p$-group of class at most two and exponent $p$, then there exist groups $G_1$ and $G_2$ of class two and exponent $p$ that contain $G$, neither of which can be expressed as a central product, and with $G_1$ capable and $G_2$ not capable. We provide upper bounds for ${\rm rank}(G_i^{\rm ab})$ in terms of ${\rm rank}(G^{\rm ab})$ in each case.
We compute the numbers g(n,2,2) of nilpotent groups of order n, of class at most 2 generated by at most 2 generators, by giving an explicit formula for the Dirichlet generating function \sum_{n=1}^\infty g(n,2,2)n^{-s}.
We study finite capable $p$-groups $G$ of nilpotency class 2 such that the commutator subgroup $γ_2(G)$ of $G$ is cyclic and the center of $G$ is contained in the Frattini subgroup of $G$.
We show that in the category of groups, every singly-generated class which is closed under isomorphisms, direct limits and extensions is also singly-generated under isomorphisms and direct limits, and in particular is co-reflective. We also establish several new relations between singly-generated closed classes.
For each family of finite classical groups, and their associated simple quotients, we provide an explicit presentation on a specific generating set of size at most 8. Since there exist efficient algorithms to construct this generating set in any copy of the group, our presentations can be used to verify claimed isomorphisms between representations of the classical group. The presentations are available in Magma.
The main result of this paper is a complete classification of the outer automorphism groups of two-generator, one-relator groups with torsion. To this classification we apply recent algorithmic results of Dahmani--Guirardel, which yields an algorithm to compute the isomorphism class of the outer automorphism group of a given two-generator, one-relator group with torsion.
A group is called capable if it is a central factor group. We consider the capability of finite groups of class two and exponent $p$, $p$ an odd prime. We restate the problem of capability as a problem about linear transformations, which may be checked explicitly for any specific instance of the problem. We use this restatement to derive some known results, and prove new ones. Among them, we reduce the general problem to an oft-considered special case, and prove that a 3-generated group of class 2 and exponent $p$ is either cyclic or capable.
We calculate explicitly the normal zeta function of the free class two nilpotent group on four generators. This has Hirsch length ten.
We provide a simple criterion for an element of the mapping class group of a closed surface to have normal closure equal to the whole mapping class group. We apply this to show that every nontrivial periodic mapping class that is not a hyperelliptic involution is a normal generator for the mapping class group when the genus is at least 3. We also give many examples of pseudo-Anosov normal generators, answering a question of D. D. Long. In fact we show that every pseudo-Anosov mapping class with stretch factor less than $\sqrt{2}$ is a normal generator. Even more, we give pseudo-Anosov normal g…
On Nielsen equivalence classes of two-elements generators of mapping class groups
本文研究闭有向曲面映射类群的二元生成元分类,证明亏格不小于八时存在无穷多个 Nielsen 等价类。
Explicit generators of the Steinberg Module of the mapping class group
本研究证明了Broaddus猜想,给出映射类群作用于Harvey曲线复形同调的唯一轨道代表元的简单刻画,并证明亏格g曲面映射类群作用于Steinberg模的核是平凡的。
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该研究探讨映射类群的正规生成问题,给出映射类成为正规生成元的条件,提出基于Teichmüller空间渐近平移长度的可约映射类正规生成判别准则,类比了Lanier-Margalit的伪Anosov生成元相关研究。
Normal generators for mapping class groups
研究测地群的正则生成元与Teichmüller空间和曲率图上渐近翻译长度的关系,探讨相关开放问题。
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