If an automatic structure exists for a group or coset system, all the information contained in the word-acceptor and the various multipliers can be stored in a single FSA. This FSA is called the general multiplier, and it has labelled accept states. If the word pair (*u,v*) is accepted then *u* and *v* are both accepted words (accepted by the word-acceptor that is), and the label for the accept state consists of a list of words. Each word in the list of words forming the label is either `IdWord`

or a word consisting of one of the generating symbols from the alphabet. A word *w* is on the list when it is true that *u*w = _{G} v*, (or, in the case of a coset system that

`IdWord`

or one of the generators. In fact, it is not hard to see that we can extend this concept further, and include any word in the label for a word pair (The existence of a word-acceptor and multipliers gives us some important information about a group, which we have already hinted at in the discussion of the general multiplier FSA. Let us consider what it means if, in the process of testing two distinct word pairs (*u,v*), and (*t,w*) for acceptance by a general multiplier, both pass through some state σ. So, we have found that we can reach state σ by two distinct routes (*u _{0}*u_{1}*...*u_{n},v_{0}*v_{1}*...*v_{n}*) and (

This means that with each state of the FSA we can associate a definite element from the group, because, *r _{0}*...*r_{k}*_{g}*S_{k}*...S_{0} =_{G} U_{n}*...*U_{0}*v_{0}*...*v_{n}* and

So given any equation between words of the form *ug = v*, we can form the series of elements, 1, *U _{0}*v_{0}*,

`IdWord`

and back. Each of these elements is called a "word-difference", and we have just shown that if the general multiplier exists there are only finitely many of them. Also, we can use the fact that there are finitely many word-differences in order to construct the word-acceptor and then the general multiplier.we shall make a few general comments about the properties of the set of word-differences:

- The set of word-differences always includes
`IdWord`

. This is obvious from the construction. - The set of word-differences always includes each generator
`g`

, except in the case where the generator is reducible. This is because our original set of generators is closed under inversion so that for every*g*,`G*g=IdWord`

is an equation, and the element*g*arises as a word-difference of this equation. - The set of word-differences is closed under inversion. This is because every word-difference by definition arises from an equation of the form
*ug =*is which_{G}v*u*and*v*have no common prefix and both are reduced. If*ug =*is an equation and_{G}v*v*is not`IdWord`

, then*vG =*must also be an equation. For_{G}u*vG*is certainly equal to*u*, and*vG > u*because if*vG =*we could cancel_{G}u*v*from the right and left of the equation*ug =*contrarary to our claim that_{G}v*u*and*v*have no common prefix. Finally*vG*is not less than*u*because if it were then it would be a reduction of*u*contrary to*u*being irreducible. To deal with case of equations of the form*ug =*: these must be those of the form_{G}Idword*Gg*=_{G}`Idword`

, and we insisted that our alphabet was itself closed under inversion.