AI Summary & Key Takeaways
AI InsightIn particular, if the all elements of G' can be represented as a commutator, then |G'|\leq |G:Z(G)|2.
📌 3 Key Takeaways
Abstract
A sufficient condition such that any element of G' (the commutator subgroup of G) can be represented as a product of n commutators, was studied in \cite{GAL62}. In this article we study a necessary and sufficient condition such that any element of G' can be represented as a product of n commutators, Let n be the smallest nature number such that any element of finite group G can be represented as a product of n commutators. A group G with this property will be called an n -commutator group, and n will be denoted by c(G) . Then \frac{\ln(|G'|)}{\ln(|G:Z(G)|)} \leq 2c(G). In particular, if the all elements of G' can be represented as a commutator, then |G'|\leq |G:Z(G)|2.
Özet
A sufficient condition such that any element of G' (the commutator subgroup of G) can be represented as a product of n commutators, was studied in \cite{GAL62}. In this article we study a necessary and sufficient condition such that any element of G' can be represented as a product of n commutators, Let n be the smallest nature number such that any element of finite group G can be represented as a product of n commutators. A group G with this property will be called an n -commutator group, and n will be denoted by c(G) . Then \frac{\ln(|G'|)}{\ln(|G:Z(G)|)} \leq 2c(G). In particular, if the all elements of G' can be represented as a commutator, then |G'|\leq |G:Z(G)|2.