International Journal of Mathematics, Game Theory and Algebra, Volume (18), No (1), Year (2008-2) , Pages (11-17)

#### Title : ( On the growth sequences of the free products of some copies of PSL(n,q )

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Let $G$ be a finitely generated group and $G^n$ be the direct product of $n$ copies of $G$. The growth sequence of $G$ is the sequence $\\\\\\\\{d(G^n)\\\\\\\\}_{n \\\\\\\\geq1}$, where $d(G^n)$ is the minimum number of generators of $G^n$. The purpose of this article is to investigate the growth sequences of $G$, where $G=\\\\\\\\kcopy$ is the free products of $k$ copies of the projective special linear simple group $\\\\\\\\ps$, $m,q \\\\\\\\geq 2$. In fact, we establish that $d\\\\\\\\left(G^{h(2,\\\\\\\\ps)^k}\\\\\\\\right) = 2k$ for all $m,q \\\\\\\\geq 2$, where $h(2,\\\\\\\\ps)$ is the maximum number $t$ such that $d(\\\\\\\\ps ^t)=2$. Moreover, we prove that \\\\\\\\\\\\\\\\ $d\\\\\\\\left(( \\\\\\\\mkcopy )^t \\\\\\\\right)=2k$ \\\\\\\\hspace{0.2 cm} for all $m_i,q_i \\\\\\\\geq 2$, $1 \\\\\\\\leq i \\\\\\\\leq k$ and \\\\\\\\hspace{0.2 cm} $1 \\\\\\\\leq t \\\\\\\\leq h(2,\\\\\\\\pk)h(2,\\\\\\\\pkk) \\\\\\\\cdots h(2,\\\\\\\\pkn)$.\\\\\\\\ \\\\\\\\noindent We have also confirmed the above results by several examples in find section.\\\\\\\\\\\\\\\\

#### Keywords

, Minimum number of generators, growth sequences, free product, projective linear
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@article{paperid:1009473,
title = {On the growth sequences of the free products of some copies of PSL(n,q},
journal = {International Journal of Mathematics, Game Theory and Algebra},
year = {2008},
volume = {18},
number = {1},
month = {February},
issn = {1060-9881},
pages = {11--17},
numpages = {6},
keywords = {Minimum number of generators; growth sequences;free product; projective linear group},
}