Efficient Presentations For PSL(2,p) and Related Groups
Originally written as part of MT4824 Topics in Groups instructed by Dr C M Cambell for the MSc in Pure Mathematics at the University of St Andrews.
Definitions
The deficiency 
of a group 
is the largest integer such that
has a presentation with 
generators and 
relations. If 
is finite, then 
is a negative number.
An efficient presentation of a finite group 
has 
relations, where 
is the rank of the Schur multiplier, which is
defined thus:
where 
is the free group on the set of generators of 
, and
is the normal closure of the relations holding in 
. The
groups 
have Schur multiplier 
, which is
cyclic, and hence of rank 1. The groups 
has
trivial Schur multiplier.
Thus an efficient presentation of 
will have 2
generators and 3 relators, whereas 
will have 2
generators and 3 relators.
Relationships between 
and 
The projective special linear groups 
are
obviously defined by
Date: 2009-06-12 22:04:26 BST
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