By Vogel H.-J.

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Since N' with s' also satisfies the Peano postulates, the same recursion diagram produces a unique homomorphism g: N' ~N in the opposite direction, as in the bottom row of the commutative diagram below: Now compare the composite function g-j: N ---N with the identity function I: N ---N. They both make the diagram 0 -N ~N I"g·! II IiII,g.!

Thus each transformation T has an inverse T- 1: X ..... X; any two transformations Sand T have a composite S· T -first apply T and then S. A transformation group G on a set X is a non-empty set G of transformations T on X which contains with each T its inverse and with any two transformations S, T in G their composite. This implies that G always contains the identity transformation I on X: (1) A transformation group on a finite set (and especially on the typical finite set (l,2, ... , n}) is usually called a permutation group.

To summarize: symmetry forces us to consider transformation groups, and even forces thoughts as to more abstractions from this notion. 8. Groups For any three transformations R, S, and T of a set X the iterated composite, by its definition, satisfies «R,S)'Dx = R(S(Tx)) = (R,(S'D)x, so composition of transformations is associative. Now, in a transformation group G, forget the fact that the elements T of G transform things, and use only the properties of composition. It is then a group in the sense of the following definition of an "abstract" group: 23 8.