**MSC:**- 17B30 Solvable, nilpotent algebras
- 20F05 Generators, relations, and presentations

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 $d$.