# Algebraic and geometric topology. Proceedings of symposia in by James R. Milgram

By James R. Milgram

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Extra info for Algebraic and geometric topology. Proceedings of symposia in pure mathematics, V.32, Part.2

Example text

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.