M. F. Newman, G. Sauerbier, J. Wisliceny
Groups of prime-power order with a small number of relations
The paper is published: Rostocker Mathematische Kolloquium, Rostock. Math. Kolloq. 49, 141-154 (1995)
17B30 Solvable, nilpotent algebras
20F05 Generators, relations, and presentations
Abstract: The Theorem of {\sc Golod/Shafarevich} (1964) (briefly GST)
gives a
lower bound for the number of relations needed to define a
group of prime-power order. Specifically let $G$ be a
group of prime-power order and $d$ the generator number of $G$,
then every presentation for $G$ has more than ${d^2/4}$
relations. (The generator number is the size of a minimal
generating set.) The GST actually says that every
pro-$p$-presentation for $G$ has more than ${d^2/4}$ relations.
The same inequality also holds for presentations of nilpotent
algebras with finite generator number $d$ ({\sc Koch} 1977). It
is natural to ask how sharp this inequality is (see for example
{\sc Kostrikin} 1965). {\sc Wisliceny} (1981) has shown that,
odd primes $p$, there are pro-$p$-presentations with $d$
generators and $d^2/4 +d/2 - (7+(-1)^d)/8$ relations which
define finite pro-$p$-groups (and also that there are
for Lie algebras with the same numbers of generators and of
which define nilpotent Lie algebras)
with generator number $d$.
Notes: Abstract contains the first few lines of text of the paper.