Products of Characters and Finite $p$-groups III

群的环积类积及其冯·诺依曼代数 III：嵌入

Wreath-like products of groups and their von Neumann algebras III: Embeddings

本文研究具有性质 (T) 的环积类积群，完整刻画了其冯·诺依曼代数间的嵌入，并由此构造出大量两两不可嵌入的 ICC 群及仅具有内自同态的群冯·诺依曼代数。

关于出现在一族图中的素数特征度数图（iii）

On prime character degree graphs occurring within a family of graphs (iii)

通过扩展构造的灵活性，完整分类了可解群的素数特征度数图中可能出现的图。

Regular orbits of finite primitive solvable groups, III

Suppose that a finite solvable group $G$ acts faithfully, irreducibly and quasi-primitively on a finite vector space $V$. Then $G$ has a uniquely determined normal subgroup $E$ which is a direct product of extraspecial $p$-groups for various $p$ and we denote $e=\sqrt{|E/\bZ(E)|}$. We prove that when $e=2,3,4,8,9,16$, $G$ will have regular orbits on $V$ when the corresponding vector space is not too small.

On the Transposition Anti-Involution in Real Clifford Algebras III: The Automorphism Group of the Transposition Scalar Product on Spinor Spaces

A signature epsilon=(p,q) dependent transposition anti-involution T of real Clifford algebras Cl_{p,q} for non-degenerate quadratic forms was introduced in [arXiv.1005.3554v1]. In [arXiv.1005.3558v1] we showed that, depending on the value of (p-q) mod 8, the map T gives rise to transposition, complex Hermitian, or quaternionic Hermitian conjugation of representation matrices in spinor representation. The resulting scalar product is in general different from the two known standard scalar products [Lounesto, Clifford algebras and Spinors 2001]. We provide a full signature (p,q) dependent classif…

On the structure of groups with polynomial growth III

We show that a compactly generated locally compact group of polynomial growth having no non-trivial compact normal subgroups can be embedded as a co-compact subgroup into a semidirect product of a connected, simply connected, nilpotent Lie group and a compact group. There is also a uniqueness statement for this extension.

Generically free representations III: extremely bad characteristic

In parts I and II, we determined which faithful irreducible representations $V$ of a simple linear algebraic group $G$ are generically free for Lie($G$), i.e., which $V$ have an open subset consisting of vectors whose stabilizer in Lie($G$) is zero, with some assumptions on the characteristic of the field. This paper settles the remaining cases, which are of a different nature because Lie($G$) has a more complicated structure and there need not exist general dimension bounds of the sort that exist in good characteristic.