This article is cited in 6 papers

Mathematical logic, algebra and number theory

On groups critical with respect to a set of natural numbers

Yu. V. Lytkin

Novosibirsk State University, Pirogova st., 2, 630090, Novosibirsk, Russia

Abstract: The spectrum of a finite group is the set of its element orders. A finite group $G$ is critical with respect to a subset $\omega$ of natural numbers, if $\omega$ is equal to the spectrum of $G$ and not equal to the spectrum of any proper section of $G$. For any natural number $n$, we construct $n$ finite critical groups with the same spectrum. We also give a complete description of finite groups critical with respect to the spectrum of the alternating group of degree 6 and the spectrum of the special linear group of dimension 3 over a field of order 3.

Keywords: finite group, spectrum, critical group.

UDC: 512.542

MSC: 20D06, 20D60

Received November 7, 2013, published December 14, 2013

Language: English