A group equation over a group 
is an equation 
with 
in the free product 
, where 
is the free group on the unknowns
, ..., 
. A solution in an overgroup substitutes elements for the
unknowns in a group containing an isomorphic
copy of 
,
with the given coefficients retained. The quotient
 |
where the denominator is the normal closure, is the universal group imposing a finite
system. The coefficient map 
is injective iff
the system has a solution in some overgroup. Solutions need not exist in 
itself.
The exponent-sum matrix has one row for each word 
and one column for each unknown. Its
entry is the total exponent of 
in 
. The system is nonsingular when this integer
matrix has full row matrix rank over 
. For example, 
is nonsingular, even when this equation has no solution
in 
.
The Howie conjecture concerns solvability of
nonsingular systems in overgroups.
