[Paper Review] Generalized power central group identities in almost subnormal subgroups of $\GL_n(D)$

Bui Xuan Hai, H. V. Khanh+1 more|arXiv (Cornell University)|Jan 18, 2018

Finite Group Theory Research17 references3 citations

TL;DR

This paper investigates generalized power central group identities (GPCGIs) in almost subnormal subgroups of $\mathrm{GL}_n(D)$, where $D$ is a division ring. It proves that if such a subgroup satisfies a non-trivial GPCGI, then it must be central—establishing a strong structural constraint on these subgroups under the given identity condition.

ABSTRACT

In this paper, we study almost subnormal subgroups of the general linear group $\GL_n(D)$ of degree $n\ge 1$ over a division ring $D$ that satisfy a generalized power central group identity.

Motivation & Objective

• To investigate the structure of almost subnormal subgroups in $\mathrm{GL}_n(D)$ that satisfy generalized power central group identities (GPCGIs).
• To determine whether such subgroups must be central when they satisfy a non-trivial GPCGI.
• To extend known results on group identities and free subgroups in skew linear groups to the setting of almost subnormal subgroups.
• To analyze the role of the center of the division ring and the algebraic properties of elements in these subgroups.
• To resolve Conjecture 1.2 by proving that non-central almost subnormal subgroups cannot satisfy non-trivial GPCGIs unless they are central.

Proposed method

• Utilizes the concept of generalized group monomials over $\mathrm{GL}_n(D)$, particularly those of the form $w = a_1 x_{i_1}^{\alpha_1} \cdots a_t x_{i_t}^{\alpha_t} a_{t+1}$ with $a_i \in \mathrm{GL}_n(D)^*$.
• Applies the notion of generalized power central group identities (GPCGIs), where $w(c_1,\dots,c_m)^p \in F$ for some $p$ depending on the tuple.
• Employs the Cartan-Brauer-Hua theorem and Amitsur’s Theorem to analyze the center and structure of subrings generated by almost subnormal subgroups.
• Uses the theory of free products and the Nielsen-Schreier Theorem to show that certain subgroups contain non-cyclic free subgroups when non-central algebraic elements exist.
• Applies results from [21] on normality of non-central almost subnormal subgroups in $\mathrm{GL}_n(D)$ for $n \geq 2$, and extends to $n=1$ via center uncountability.
• Leverages the existence of non-cyclic free subgroups in $D^*$ when the center is uncountable and contains non-central algebraic elements, as established in [12] and [6].

Experimental results

Research questions

• RQ1 Under what conditions can a non-central almost subnormal subgroup of $\mathrm{GL}_n(D)$ satisfy a non-trivial generalized power central group identity?
• RQ2 Is it possible for a non-central almost subnormal subgroup of $\mathrm{GL}_n(D)$ to satisfy a GPCGI without being central?
• RQ3 How does the uncountability of the center of $D$ affect the existence of free subgroups and the validity of GPCGIs?
• RQ4 What structural constraints arise in almost subnormal subgroups when they satisfy a GPCGI, especially in the case $n=1$?
• RQ5 To what extent do results on free subgroups in $D^*$ extend to almost subnormal subgroups of $D^*$?

Key findings

• If a non-central almost subnormal subgroup $N$ of $\mathrm{GL}_n(D)$ satisfies a non-trivial GPCGI, then $N$ must be central, thus confirming Conjecture 1.2.
• For $n \geq 2$, any non-central almost subnormal subgroup of $\mathrm{GL}_n(D)$ contains $\mathrm{SL}_n(D)$, and its subring $F[N]$ is equal to $\mathrm{M}_n(D)$, leading to $Z(N) \subseteq F$.
• When $n=1$ and the center $F$ is infinite, the center of a non-central almost subnormal subgroup $H$ of $D^*$ is still contained in $F$.
• If $D$ has an uncountable center and $G$ is a non-central almost subnormal subgroup of $D^*$ containing a non-central algebraic element over $F$, then $G$ contains a non-cyclic free subgroup.
• In the case of infinite order algebraic elements, the existence of a free product $\langle u \rangle * \langle v \rangle$ within $G$ ensures the existence of non-abelian free subgroups via the Nielsen-Schreier Theorem.
• The proof relies on constructing elements $x = c_r(u,v)$ and $y = c_r(u,v^2)$ in $G$ that generate a non-abelian free group, thus establishing the presence of a non-cyclic free subgroup.