
\titleofpresentation{Large characteristic
subgroups and their applications to almost regular automorphisms}

\presenter{Natalia Makarenko}  %  Name just one speaker, please; collaborators
            %  can be credited in the body of the abstract.

%  [Body of abstract goes here.]
I present our recent joint results with E.~I.~Khukhro, which enable
us to solve certain important problems concerning groups with almost
regular automorphisms.

In the theory of groups with almost regular automorphisms the
final aim is often to prove the existence of a nilpotent subgroup
with bounds for the nilpotency class and index. But in many
situations we can construct such a nilpotent subgroup, say $K$, of
bounded class $c$ and bounded index $n$ only in some normal
subgroup, say $H$, and not in the group $G$ itself. To use
induction we need to consider the quotient $G/K$ and therefore it
is important to have $K$ to be characteristic in $H$. Sometimes we
can consider the automorphic closure $\prod_{\alpha\in {\rm
Aut\,}H}K^{\alpha}$, which is a characteristic nilpotent subgroup
of class $\leq cn$. However in an induction on the length of a
certain subnormal series it is desirable not to increase the
nilpotency class of the subgroup at each step. We show that if a
group has a nilpotent subgroup of class $c$ and index $n$, then
there is a characteristic nilpotent subgroup of the same class $
c$ whose index is bounded in terms of $n$ and~$c$. Moreover, we
obtain the analogous result for a subgroup satisfying an arbitrary
multilinear commutator identity. We also prove similar theorems
for a nilpotent normal subgroup (or an ideal in a Lie algebra)
with a bound for the rank (dimension) of the quotient group
(algebra).


We now give  precise formulations. We say for short that a certain
quantity is $(a,b,\ldots )$-bounded if it is bounded above by some
function depending only on $a,b,\ldots $ Let $x_1,\,x_2,\ldots $
be group variables.

\begin{definition} Multilinear commutators of weight $1$ in the
variables $x_i$ are the variables $x_i$ themselves. By induction,
\emph{multilinear commutators of weight $w>1$ in the variables
$x_i$} are commutators of the form $\varkappa =[\varkappa
_1,\,\varkappa _2]$, where $\varkappa _1$ and $\varkappa _2$ are
multilinear commutators with disjoint sets of variables of weights
$w_1$ and $w_2$ for $w=w_1+w_2$.
\end{definition}

A group $G$ satisfies a multilinear commutator identity $\varkappa
=1$ if and only if the commutator (verbal) subgroup $\varkappa
(G)$ obtained by replacing all the variables in the commutator
$\varkappa$ by the group $G$ is trivial.

\begin{theorem} \label{makarenkoth1}  Let $\varkappa$ be a multilinear commutator of
weight $w$. If a group $G$ contains a subgroup $H$ of finite index
$|G:H|=n$ satisfying the identity $\varkappa (H)=1$, then $G$ also
contains a characteristic subgroup $C$ satisfying the identity
$\varkappa (C)=1$ whose index is finite and $(n,w)$-bounded.
\end{theorem}


Such a result was so far known in folklore only for abelian
subgroups, that is, for $\varkappa = [x_1, x_2]$.

We say that an ideal $J$ of a Lie algebra $L$ is \emph{
automorphically-invariant\/} if $J$ is invariant under all
automorphisms of~$L$. A Lie algebra $L$ is \emph{nilpotent of
class $\leq c$} if $\gamma _{c+1}(L):=[\underbrace{L,\ldots,
L}_{c+1}\,]=0$.

\begin{theorem} If a Lie algebra  $L$ over any field
has a nilpotent ideal of nilpotency class $c$ and of finite
codimension $r$, then $L$ has also an automorphically-invariant
ideal of finite $(r,c)$-bounded codimension that is nilpotent of
class $\leq c$.
\end{theorem}

This result, as well as ideas in its proof, are used to prove a
similar result for groups, where the role of dimension is played
by rank.

Recall that a group has rank $r$ if every finitely generated
subgroup of it can be generated by $r$ elements.



\begin{theorem}\label{makarenkoth3} Suppose that a group $G$ has a nilpotent normal subgroup $H$ of
nilpotency class $c$ such that the quotient group $G/H$ has finite
rank $r$. If either
\begin{enumerate}
\item
$H$ is periodic, or
\item
$H$ is torsion-free,
\end{enumerate}
then $G$ has also a characteristic nilpotent subgroup $C$ of
nilpotency class $\leq c$ with quotient group $G/C$ of finite $
(r, c)$-bounded rank. 
\end{theorem}

We apply the above theorems for obtaining some results on almost
regular automorphisms, which had previously stalled precisely
because of the need for converting subnormal series into normal.
Theorem \ref{makarenkoth1} proved to be useful in the study of
almost regular automorphisms of order~4.



\begin{theorem} There exist a constant $c$ and a function of a positive
integer argument~$f(m)$ such that if a finite group $G$ admits an
automorphism of order~$4$ with exactly $m$ fixed points, then $G$
has characteristic subgroups $N\leq H\leq G$ such that $|G/H|\leq
f(m)$, the quotient group $H/N$ is nilpotent of class~$\leq 2$,
and the subgroup $N$ is nilpotent of class~$\leq c$.
\end{theorem}

This result gives an affirmative answer to Shumyatsky's question
11.126 in the ``Kourovka Notebook''.

Theorem \ref{makarenkoth3} is applied to automorphisms of prime
order that are almost regular in the sense of small rank of the
fixed-point subgroup. In \cite[Theorem 2]{makarenko1} it was
proved that if a finite nilpotent group $G$ of derived length $d$
admits an automorphism of prime order $p$ with centralizer of
rank~$r$, then $G$ has a subnormal series of $(p,r,d)$-bounded
length whose smallest term is nilpotent of $p$-bounded class,
while the other factors have $(p,r,d)$-bounded rank. Now this
result can be strengthened. Using also the result in
\cite{makarenko2}, where Hall--Higman type theorems were combined
with powerful $p$-groups to obtain almost nilpotency of a finite
group with such an automorphism, we now have the following
corollary.

\begin{corollary*} If a finite soluble group $G$ of derived length $d$ admits an
automorphism of prime order $p$ with centralizer of rank~$r$, then
$G$ has characteristic subgroups $R\leq C\leq G$ such that the
quotient group $C/R$ is nilpotent of $p$-bounded class, the
subgroup $R$ has $(p,r)$-bounded rank, and the quotient $G/C$ has
$(p,r,d)$-bounded rank; if $G$ is nilpotent, then one can put
$R=1$.
\end{corollary*}



\begin{ourbiblio}{99}  %  Delete this environment if you
               %  aren't giving a bibliography.

\bibitem{makarenko1}
{\sc E.\,I.~Khukhro}, Finite soluble and nilpotent groups with a
restriction on the rank of the centralizer of an automorphism of
prime order,
 \emph{Sibirsk. Mat.~Zh.}, {\bf 41} (2000), 451--469; Engl. transl.,
\emph{Siberian Math.~J.}, {\bf 41} (2000), 373--388.

\bibitem{makarenko2}  {\sc E.\,I.~Khukhro and V.\,D.~Mazurov},
Finite groups with an automorphism of prime order whose
centralizer has small rank, \emph{J. Algebra}, {\bf 301} (2006),
474--492.

\end{ourbiblio}

\presenteraddress{Institute of Mathematics, Novosibirsk}  % The name of a university is fine.

\email{makarenk@math.nsc.ru}
