\documentclass{article}
\usepackage{amsmath}
\setcounter{MaxMatrixCols}{30}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{graphicx}
\begin{document}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newenvironment{proof}[1][Proof]{\noindent\textbf{#1.} }{\ \rule{0.5em}{0.5em}}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{axiom}[theorem]{Axiom}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{solution}[theorem]{Solution}
\newtheorem{summary}[theorem]{Summary}
\newtheorem{criterion}[theorem]{Criterion}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{case}[theorem]{Case}
\newtheorem{claim}[theorem]{Claim}
\newtheorem{conclusion}[theorem]{Conclusion}


\title{Generalized PC-Groups And Generalized CC-Groups Of Matrices}
\date{}
\maketitle
\begin{center}

\author {\textbf{Francesco Russo}}

\end{center}








\vspace{1cm}
\begin{center}
\textbf{ABSTRACT}
\end{center}




Let $\mathfrak{X}$ be a class of groups. An element $x$ of a group
$G$ is said to be $\mathfrak{X}C$-$central$, or an $XC$-$element$,
if $G/C_G(x^G)$ satisfies $\mathfrak{X}$, where the symbol $x^G$
denotes the normal closure of the subgroup $<x>$ in $G$ (see
\cite{vI}). A $\mathfrak{X}C$-central group $G$ is also called $a$
$group$ $with$ $\mathfrak{X}$-$conjugacy$ $classes$. An $XC$-element
$x$ of a group $G$ is also called $an$ $element$ $which$ $has$
$\mathfrak{X}$-$conjugacy$ $classes$. The factor group $G/C_G(x^G)$
is a group of automorphisms of $x^G$ and an old question of Group
Theory is the knowledge of a group, starting from information on its
group of automorphisms.

Following \cite{vI}, a class $\mathfrak{X}$ of groups is said to be
$\mathbf{S}_n$-$closed$ if every normal subgroup of an $\mathfrak
{X}$-group is an $\mathfrak {X}$-group. A class $\mathfrak{X}$ of
groups is said to be $\mathbf{N}$-$closed$ if the subgroup generated
by any system of normal $\mathfrak{X}$-subgroups of a group is an
$\mathfrak {X}$-group. A class $\mathfrak{X}$ of groups is said to
be $\mathbf{P}$-$closed$ if every extension of an $\mathfrak
{X}$-group is an $\mathfrak {X}$-group. From now the symbol
$\mathfrak{X}$ will denote a class of groups which is
$\mathbf{S}_n$, $\mathbf{N}$ and $\mathbf{P}$-closed.


It is possible to introduce some series which generalize the usual
notion of upper central series of a group (see for instance [10,
Chapter 2]) thanks to the following remark.

Let $G$ be a group. If $x$ and $y$ are $XC$-elements of $G$, then
both $G/C_G(x^G)$ and $G/C_G(y^G)$ satisfies $\mathfrak{X}$, so
$G/(C_G(x^G)\cap C_G(y^G))$ satisfies $\mathfrak{X}$. But the
intersection of $C_G(x^G)$ with $C_G(y^G)$ lies in
$C_G((xy^{-1})^G)$, so that $G/C_G((xy^{-1})^G)$ satisfies
$\mathfrak{X}$ and $xy^{-1}$ is an $XC$-element of $G$. Hence the
$XC$-elements of $G$ form a subgroup $X(G)$ and it is easy to see
that $X(G)$ is characteristic in $G$.

This remark has been first noted by R.Baer for the class of
$FC$-groups (see [10, Lemma 4.31]) and allows us to define the
series

$$1=X_0 \triangleleft{X_1}\triangleleft \ldots \triangleleft{X_{\alpha}}\triangleleft{X_{\alpha + 1}}\triangleleft \ldots,$$
where $X_1 = X(G)$, $X_{\alpha + 1}/X_{\alpha}=X(G/X_{\alpha})$ and
$$X_{\lambda} = \bigcup_{\alpha < \lambda} X_{\alpha},$$
with $\alpha$ ordinal and $\lambda$ limit ordinal. This series is a
characteristic ascending series of $G$ and  it is called $upper$
$XC$-$central\,\, series$ of $G$. The last term of the upper
$XC$-central series of $G$ is called $XC$-$hypercenter$ of $G$. If
$G = X_{\beta}$, for some ordinal $\beta$, we say that $G$ is an
$XC$-$hypercentral$ group of type at most $\beta$.





The $XC$-$length$ of an $XC$-hypercentral group is defined to be the
least ordinal $\beta$ such that $G=X_{\beta}$, in particular when $G
= X_c$ for some positive integer $c$, we say that $G$ is
$XC$-$nilpotent$ of length $c$.  A group $G$ is said to be an
$XC$-$group$ if all its elements are $XC$-elements, that is, if $G$
has $XC$-length at most 1. The first term $X(G)$ of the upper
$XC$-central series of $G$ is called $XC$-$center$ of $G$ and the
$\alpha$-th term of the upper $XC$-central series of $G$ is called
$XC$-$center$ $of$ $length$ $\alpha$ of $G$. Roughly speaking, the
upper $XC$-central series of $G$ measures the distance of $G$ to be
an $XC$-group.


If $\mathfrak {X}$ is the class $\mathfrak{F}$ of finite groups, we
have the notion of $FC$-hypercentral group and classical results can
be found in [10, Chapter 4]. In this circumstance, we specialize the
previous symbol $X$ in the symbol $F$, for introducing
$XC$-hypercentral series and related definitions. If $\mathfrak {X}$
is the class $\mathfrak{P}\mathfrak{F}$ of polycyclic-by-finite
groups, we have the notion of $PC$-hypercentral group and a complete
description of such groups can be found in \cite{pc}. In this
circumstance, we specialize the previous symbol $X$ in the symbol
$P$, for introducing $PC$-hypercentral series and related
definitions. If $\mathfrak {X}$ is the class $\mathfrak {C}$ of
Chernikov groups, we have the notion of $CC$-hypercentral group and
a complete description of such groups can be found in \cite{cc1}. In
this circumstance, we specialize the previous symbol $X$ in the
symbol $C$, for introducing $CC$-hypercentral series and related
definitions.

Our approach to generalized central series of a group has been
introduced by D.H.McLain, obtaining classical results in Theory of
Generalized $FC$-groups (see [10, Theorem 4.37, Theorem 4.38]). Also
\cite{gar} obtain similar results with this approach. One of the
most recent improvement of McLain Theorems has been furnished by [1,
Theorem A, Theorem B, Theorem C] for $PC$-hypercentral groups and
$CC$-hypercentral groups.

On the other hand hypercentral groups, $FC$-hypercentral groups,
$PC$- hypercentral groups and $CC$-hypercentral groups can be much
different between themselves as it is shown either by the
consideration of the infinite dihedral group or by means of examples
in \cite{bgm} and \cite{tl}.


Our results are devoted to groups of matrices of $PC$-hypercentral
groups and $CC$-hypercentral groups. A group $G$ is called $linear$
$of$ $degree$ $n$ $over$ $an$ $arbitrary$ $field$ $K$, or a group of
$n\times n$ matrices over an arbitrary field $K$, or briefly
$linear$, if $G$ is a subgroup of the group $GL(n,K)$, where $n$ is
a positive integer. Classical literature on linear groups is
represented by \cite{merz}, \cite{pla}, \cite{sup}, \cite{W1}. More
recent is \cite{pin}, where [8, Chapter 5] gives a survey on chains
conditions in linear groups. These chains conditions have been first
introduced in Russian literature (see \cite{merz}, \cite{pla},
\cite{sup}). We prove the following results, which extend [7,
Theorem 2].


 \vspace{2cm} \textbf{Theorem A.} \textit{Let $G$ be a
linear group of degree $n$ over an arbitrary field $K$. Then the
following conditions are equivalent
\begin{itemize}
\item [(i)]   $G$ is $PC$-nilpotent;
\item [(ii)]  $G$ is $PC$-hypercentral;
\item [(iii)] $G$ is nilpotent-by-polycyclic-by-finite.
\end{itemize}}}


 \vspace{0.5cm} \textbf{Theorem B.} \textit{Let $G$ be a
linear group of degree $n$ over an arbitrary field $K$. If $charK=0$
and $G/Z(G)$ does not contain subgroups of infinite exponent, then
the following conditions are equivalent
\begin{itemize}
\item [(i)]   $G$ is $FC$-nilpotent;
\item [(ii)]  $G$ is $FC$-hypercentral;
\item [(iii)] $G$ is nilpotent-by-finite;
\item [(iv)]  $G$ is $CC$-nilpotent;
\item [(v)]   $G$ is $CC$-hypercentral.
\end{itemize}}

\vspace{0.3cm}
\\
The construction in [11] will allow us to give examples of groups
which satisfy Theorem A and Theorem B.

\small \vspace{0.3cm}

\textit{Key Words}: Conjugacy classes; linear $PC$-groups; linear
$CC$-groups; $PC$-hypercentral series; $CC$-hypercentral series.

\textit{MSC 2000}: 20C07; 20D10; 20F24.



\small
\begin{thebibliography}{20}

\bibitem{bgm} Beidleman, J.C., Galoppo, A. and Manfredino, M. (1998). On
$PC$-hypercentral and $CC$-hypercentral groups. \textit{Comm. Alg.}
\textbf{26}, 3045-3055.

\bibitem{pc} Franciosi, S., de Giovanni, F. and Tomkinson, M. J.(1990). Groups
with polycyclic-by-finite conjugacy classes. \textit{Boll.UMI}
\textbf{4B}, 35-55.

\bibitem{cc1} Franciosi, S., de Giovanni, F. and Tomkinson, M. J.(1991). Groups with
Chernikov conjugacy classes. \textit{J. Austral. Math. Soc} \textbf
A50}, 1-14.

\bibitem{gar} Gara\u{s}\u{c}uk, M. S.(1960). On theory of generalized nilpotent linear groups. \textit{Dokl. Akad. Nauk BSSR} [in Russian] \textbf{4} 276-277.

\bibitem{tl} Landolfi, T. (1995). On generalized central series of
groups. \textit{Ricerche Mat.} \textbf{XLIV}, 337-347.

\bibitem{merz} Merzlyakov,Yu.I. Linear groups. In:
\textit{Itogi Nauki i Tkhniki, Algebra Topologiya, Geometriya},
vol.16, 1978, pp.35-89.

\bibitem{m} Murach,M.M. (1976). Some generalized FC groups of matrices.
\textit{Ukr. Math. Journal} \textbf{28}, 92-97.

\bibitem{pin} Pinnock,C.J.E. Supersolubility and finitary groups. Ph.D. Thesis. University of London, Queen Mary and Westfield College, 2000, London.

\bibitem{pla} Platonov,V.P.(1967). Linear groups with identical relations. \textit{Dokl.Akad. Nauk BSSR} [in Russian] \textbf{11} 581-582.

\bibitem{vI} Robinson,D.J.S. \textit{Finiteness conditions and generalized
soluble groups}, Springer-Verlag, 1972, Berlin.

\bibitem {jnp} Robinson,D.J.S. and Wilson,J. (1984). Soluble groups with many polycyclic quotients. \textit{Proc. London Math.Soc.} \textbf{48},  193-229.

\bibitem{sup} Suprunenko,D.A. \textit{Groups of Matrices.} [in Russian]
Nauka, 1972, Moscow.

\bibitem{W1} Wehrfritz,B. A. F.  \textit{Infnite linear groups}. Springer, 1973, Berlin.
\end{thebibliography}




\vspace{0.2cm}  Mathematics Department, University of Naples

Office n.17, via Cinthia 80126, Naples, Italy

Phone: +39-081-675682

Fax: +39-081-7662106

E-mail: francesco.russo@dma.unina.it

\end{document}
