Computational Algebra Seminar: Yadav -- Central factor and commutator subgroup of nilpotent groups

Date: 29/08/13
Location: Carslaw 535
Time: 15:00 
Name: Manoj Yadav 
Affiliation: Harish-Chandra Research Institute 
Title: Central factor and commutator subgroup of nilpotent groups 

Abstract: Let $G$ be an arbitrary group such that $G/Z(G)$ is finite, where $Z(G)$ 
denotes the center of the group $G$.  Then $\gamma_2(G)$, the commutator subgroup 
of $G$, is finite.  This result is known as Schur’s theorem.  The converse ofthis 
result is not true in general as shown by infinite extra-special $p$-groups for odd
primes.  But when $G$ is finitely generated by $d$ elements (say) and $\gamma_2(G)$ is
finite, it was proved by B.  H.  Neumann that $|G/Z(G)| \le |\gamma_2(G)|^d$.  In this
talk I start with this basic result, discuss a slight generalization and classify all
non-abelian finite nilpotent groups $G$ minimally generated by $d := d(G)$ elements such
that central quotient is maximal, i.e., $|G/Z(G)| = |\gamma_2(G)|^d$.  First I reduce
the problem to finite $p$-groups and then notice that such $p$-groups admit maximum
number of (conjugacy) class-preserving automorphisms.  Using this and some basic Lie
theoretic techniques, I show that $d(G)$ is even.  Moreover, when the nilpotency class
of $G$ is at least $3$, then $d(G) = 2$.  The situation is then divided into two cases,
depending on whether $p$ is even (Magma is extremely useful in this case) or odd, and a
complete classification is presented.  All $p$-groups $G$ considered above satisfy the
following condition: $|x^G| = |\gamma_2(G)|$ for all $x \in G - \Phi(G)$, where $x^G$
denotes the conjugacy class of $x$ in $G$ and $\Phi(G)$ denotes the Frattini subgroup of
$G$.  Let me call such a group Camina-type.  One of my motives to visit the
Computational Algebra group here is to find a solution to the following problem: Does
there exist a finite Camina-type $p$-group $G$, $p$ odd, of nilpotency class at least
$3$ such that $\gamma_2(G) < \Phi(G)$ and $\gamma_2(G)$ is non-cyclic?