Turkish Journal of Mathematics
DOI
10.3906/mat-1406-31
Abstract
Let $\{\theta_n\}_{n=1}^\infty$ be a sequence of words. If there exists a positive integer $n$ such that $\theta_m(G)=1$ for every $m\geq n$, then we say that $G$ satisfies (*) and denote the class of all groups satisfying (*) by $\mathfrak{X}_{\{\theta_n\}_{n=1}^\infty}$. If for every proper subgroup $K$ of $G$, $K\in \mathfrak{X}_{\{\theta_n\}_{n=1}^\infty}$ but $G\notin\mathfrak{X}_{\{\theta_n\}_{n=1}^\infty}$, then we call $G$ a minimal non-$\mathfrak{X}_{\{\theta_n\}_{n=1}^\infty}$-group. Assume that $G$ is an infinite locally finite group with trivial center and $\theta_i(G)=G$ for all $i\geq 1$. In this case we mainly prove that there exists a positive integer $t$ such that for every proper normal subgroup $N$ of $G$, either $\theta_t(N)=1$ or $\theta_t(C_G(N))=1$. We also give certain useful applications of the main result.
Keywords
Locally finite groups, soluble groups, nilpotent groups, sequence of words, outer commutator words
First Page
564
Last Page
569
Recommended Citation
ARIKAN, AHMET and ERCAN, SELAMİ
(2015)
"On certain minimal non-$\mathfrak{Y}$-groups for some classes $\mathfrak{Y}$,"
Turkish Journal of Mathematics: Vol. 39:
No.
4, Article 12.
https://doi.org/10.3906/mat-1406-31
Available at:
https://journals.tubitak.gov.tr/math/vol39/iss4/12
