Groups defined by as simple as possible rules... as computer strings?

Hi,

I've just learned about group theory and I was wondering if it is possible
to define specific groups by a few equations only.

For example a simple cyclic group could be defined by g*g*g*g=1 and all
other elements just can be written as g*g and g*g*g without giving them
explicit names.

Of course sometimes these defining equations contain more than one element
and it would be good to minimize the number of elements used.

Is there a method to do this systematically and is it possible to use
equations of the form "...=1" only? (say a*b*c=1 and b*b=1)

I'd find it instructive to imagine groups as computer strings (e.g. "gg",
"ggg") which can be catenated with rules of cancellation (e.g.
"gg"+"ggg"="ggggg"="g" since "gggg"=""). In this model the associative
property is implicit.

A problem that *shouldn't* appear in the rules is that one had to expand a
string to reduce it again, e.g.
reduce "abc" with "bc=aac", "aaa="":
so that "abcd"="aaacc"="cc"

Is that possible?

Anton
.