n-Commutator Groups

Authors: A. A. MEHRVARZ, K. AZIZI

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}.

Keywords: Commutator subgroup, irreducible characters.

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