Year: 2012
Type: preprint
Abstract: Let $G$ be an arbitrary group such that $G/\Z(G)$ is finite, where $\Z(G)$ denotes the center of the group $G$. Then $γ_2(G)$, the commutator subgroup of $G$, is finite. This result is known as Shur's theorem (the Schur's theorem). In this short note we provide a quick survey on the converse of Schur's theorem, generalize known results in this direction and prove the following result (which is perhaps the most suitable statement for converse of the Schur's theorem): If $G$ is an arbitrary group with finite $γ_2(G)$, then $G/\Z(G)$ is finite if $\Z_2(G)/\Z(\Z_2(G))$ is finitely generated, where $\Z_2(G)$ denotes the second center of a group $G$. If $G/\Z(G)$ is finite, then $γ_2(G)$ is also finite and $|G/\Z(G)| \le |γ_2(G)|^d$, where $d$ denotes the number of elements in any minimal generating ser for $G/\Z(G)$. We classify all nilpotent groups $G$ of class 2 upto isoclinism (in the sense of P. Hall) such that $|G/\Z(G)| = |γ_2(G)|^d$, and ask some questions in the sequel.
Source: arXiv (Cornell University)
Author: Yadav, Manoj K.
Language: English
Topic: Finite Group Theory Research
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2277068588