By Karl Fink

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**Extra resources for A brief history of mathematics;: An authorized translation of Dr. Karl Fink's Geschichte der Elementar-Mathematik,**

**Example text**

Rhodes appeared in a book edited by M. A. Arbib [1968]. It is also presented in an elegant form by S. Eilenberg [1976]. A nice book on the generating systems of the finite symmetric groups is one by S. Picard [1946]. 4 is due to E. Galois [1832]. 5 is folklore. 7) and the observation on its relation to time reversal in the following remark, although elementary, appear to be new. 8 is well known in the literature; one can find various statements of the same flavor, although we have not seen a formulation suitable for a direct reference.

N — 1}) and the transposition y (l) = 2, Y2(2) = 1, y (i) = i for 2 < i n - 1 with respect to 1 , . . , n. Let us assume that every vertex i is covered by a coin ci, i = 1 , . . , n , and perform the following procedure: Change the coin cn of the vertex n for a copy of c n - 1 . , n. , c n - 1 ). It is clear that all steps of our procedure are allowed and that the generated transformation is also allowed. Formally, consider the mappings F ( l , . . , n) = (1, 2 , . . , n - 1, n - 1) [collapsing n to n - 1], F ( l , .

6, the (strongly connected) subdigraph D' C of D is penultimately permutation complete, a contradiction with the choice of D'. 11. A digraph D with n > 3 vertices is penultimately permutation complete if and only if it is strongly connected and contains a branch. Proof. For the necessity, first we suppose that D is not strongly connected. Then there exists a pair i,j,i j, of vertices such that there is no walk from i to j. , P(n) whenever p(j) = i and P is a permutation of the vertices such that P(n) {i, j}.

### A brief history of mathematics;: An authorized translation of Dr. Karl Fink's Geschichte der Elementar-Mathematik, by Karl Fink

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