## Abstract

So far, we have developed the concept of a group and we have seen how a great many naturally arising mathematical structures are groups. We have seen how a finite group can be expressed by means of its Cayley table, and we have seen how the concept of isomorphism allows us to say when two groups are ‘effectively equal’. What this has achieved is this: we have begun to set up a *language* of groups. It is true that this language is rather complicated, and perhaps not easy to grasp at first. That, alas, is the nature of languages. The problem is that a language needs enough flexibility to allow a good variety of sentences. With group theory, at least the scope of the language is well defined, so the language is simpler than (say) French. (There are no irregular verbs in group theory, for example.)

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