1 results
A NILPOTENCY CRITERION FOR SOME VERBAL SUBGROUPS
- Part of
-
- Journal:
- Bulletin of the Australian Mathematical Society / Volume 100 / Issue 2 / October 2019
- Published online by Cambridge University Press:
- 27 February 2019, pp. 281-289
- Print publication:
- October 2019
-
- Article
-
- You have access
- Export citation
-
The word
$w=[x_{i_{1}},x_{i_{2}},\ldots ,x_{i_{k}}]$ is a simple commutator word if 
$k\geq 2,i_{1}\neq i_{2}$ and 
$i_{j}\in \{1,\ldots ,m\}$ for some 
$m>1$. For a finite group 
$G$, we prove that if 
$i_{1}\neq i_{j}$ for every 
$j\neq 1$, then the verbal subgroup corresponding to 
$w$ is nilpotent if and only if 
$|ab|=|a||b|$ for any 
$w$-values 
$a,b\in G$ of coprime orders. We also extend the result to a residually finite group 
$G$, provided that the set of all 
$w$-values in 
$G$ is finite.
