有限群不可解性的一个判据及简单群直平方的识别

IF 0.4 3区数学 Q4 LOGIC

Algebra and Logic Pub Date : 2023-04-05 DOI:10.1007/s10469-023-09697-z

Zh. Wang, A. V. Vasil’ev, M. A. Grechkoseeva, A. Kh. Zhurtov

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引用次数: 1

摘要

有限群G的谱ω（G）是其元素的阶数集。证明了以下不可解性的充分准则：如果在群G阶的素数中，存在四个不同的素数，使得ω（G）包含它们的所有成对乘积，但不包含其中任何三个数的乘积，则G是不可解的。利用这一结果，我们证明了对于q⩾8和q≠32，简单例外Suzuki群Sz（q）的直接平方Sz（q）×Sz（qq）在有限群类中具有唯一的谱特征，而对于Sz（32）×Sz32，恰好有四个有限群具有相同的谱。

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A Criterion for Nonsolvability of a Finite Group and Recognition of Direct Squares of Simple Groups

The spectrum ω(G) of a finite group G is the set of orders of its elements. The following sufficient criterion of nonsolvability is proved: if, among the prime divisors of the order of a group G, there are four different primes such that ω(G) contains all their pairwise products but not a product of any three of these numbers, then G is nonsolvable. Using this result, we show that for q ⩾ 8 and q ≠ 32, the direct square Sz(q) × Sz(q) of the simple exceptional Suzuki group Sz(q) is uniquely characterized by its spectrum in the class of finite groups, while for Sz(32) × Sz(32), there are exactly four finite groups with the same spectrum.

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来源期刊

Algebra and Logic 数学-数学

CiteScore

1.10

自引率

20.00%

发文量

审稿时长

>12 weeks

期刊介绍： This bimonthly journal publishes results of the latest research in the areas of modern general algebra and of logic considered primarily from an algebraic viewpoint. The algebraic papers, constituting the major part of the contents, are concerned with studies in such fields as ordered, almost torsion-free, nilpotent, and metabelian groups; isomorphism rings; Lie algebras; Frattini subgroups; and clusters of algebras. In the area of logic, the periodical covers such topics as hierarchical sets, logical automata, and recursive functions. Algebra and Logic is a translation of ALGEBRA I LOGIKA, a publication of the Siberian Fund for Algebra and Logic and the Institute of Mathematics of the Siberian Branch of the Russian Academy of Sciences. All articles are peer-reviewed.