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)
MSC:
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
Lie
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,
for
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
presentations
for Lie algebras with the same numbers of generators and of
relations
which define nilpotent Lie algebras)
with generator number \$d\$.
Notes: Abstract contains the first few lines of text of the paper.