A note of the converse of Schur's Theorem

Details A note of the converse of Schur's Theorem A note of the converse of Schur's Theorem

2010-11-09

arxiv.org/abs/1011.2083v2

Mathematics

Group Theory

4 pages

Scientific paper

Let $G$ be an arbitrary group such that $G/\Z(G)$ is finite, where $\Z(G)$ denotes the center of the group $G$. Then $\gamma_2(G)$, the commutator subgroup of $G$, is finite. This result is known as Shur's theorem. The motive of this short note is to provide a quick survey on the converse of Schur's theorem and to give some further remarks. Let $\Z_2(G)$ denote the second center of a group $G$. Then we point out that a converse of Schur's theorem can be formutlated as follows: If $\gamma_2(G)$ is finite and $\Z_2(G)/\Z(G)$ is finitely generated, then $G/\Z(G)$ is finite. Moreover, $G$ is isoclinic (in the sense of P. Hall) to a finite group.

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