Conditions for the Canonical Projection to be an Isomorphism

The following theorem is the main result of this section. It provides us with combinatorial conditions that are equivalent to A being an isomorphism. 

Theorem 14.14. LetC he a finite acyclic category and letG he a finite group acting on it. The following two conditions are equivalent  

For language convenience, if one of Conditions (Cl) and (C2) is satisfied, we shall also say that Condition (C) is satisfied. In this case, the map A is an isomorphism. 

Proof of Theorem 14.14. Condition (Cl) is equivalent to requiring that G mi. mt) = G(mi. mt i, G(mt)), where this notation is used, as before, to describe all sequences m, ... g mt)) that are composable 

Example 14-15. We give an example of a group action that satisfies Condition (C) for t = 1. fc but does not satisfy Condition (C) for t = fc + 1. Let be the order sum of fc + 1 copies of the 2-element antichain. The automorphism group of Pfc+i is the direct product of fc - -1 copies of Z2. Take G to be the subgroup of index 2 consisting of elements with an even number of nonidentity terms in the product. 

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