IJPAM: Volume 28, No. 3 (2006)

NEAR FRATTINI SUBGROUPS OF CERTAIN
GENERALIZED FREE PRODUCTS OF GROUPS

Mohammad K. Azarian
Department of Mathematics
University of Evansville
1800 Lincoln Avenue
Evansville, IN 47722, USA
e-mail: azarian@evansville.edu


Abstract.Let $G=A\ast _{H}B$ be the generalized free product of the groups $A$ and $B$ with the amalgamated subgroup $H$. Also, let $\lambda (G)$ and $\psi (G)$ represent the lower near Frattini subgroup of $G$ and the near Frattini subgroup of $G$ respectively. We show that $G$ is $\psi -$free provided: $(i)$ $G$ is any ordinary free product of groups; $(ii)$ $G=A\ast _{H}B$ and there exists an element $c$ in $G\backslash H$ such that$ H^{c}\cap H=1$; $(iii)$ $G=A\ast _{H}B$ and $\lambda (G)\cap H=\mu (G)\cap H=1$; $(iv)$ $G=A\ast _{H}B$, where $A$ and $B$ are finitely generated and $\lambda -$free, and $H=C(\infty )$; $(v)$ $G=A\ast _{H}B$, and $H\neq 1$ is malnormal in at least one of $% A$ or $B$; $(vi)$ $G$ is a surface group; $(vii)$ $G$ is the group of an unknotted circle in $\mathbb{R}^{3}$; $(viii)$ $G$ is a group of $F-$type with only odd torsion where neither $U$ nor $V$ is a proper power; $(ix)$ $G$ is a non-elementary planar discontinuous group with only odd torsion. Furthermore, we show that if $G=A\ast _{H}B$, then: $(i)$ $\lambda (G)\leq H,$ provided both $A$ and $B$ are nilpotent; $(ii)$ $\psi (G)\leq H$, provided both $A$ and $B$ are finitely generated and nilpotent.

Received: April 10, 2006

AMS Subject Classification:

Key Words and Phrases: amalgamated subgroup, Frattini subgroup, Fuchsian group, generalized free product of groups, group of $F-$type, malnormal subgroup, non-elementary planar discontinuous group, near Frattini subgroup, lower near Frattini subgroup, upper near Frattini subgroup, nearly maximal subgroup, near generator, nilpotent group, non-near generator, surface group, unknotted circle

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2006
Volume: 28
Issue: 3