@MISC{Shapiro_proposition, author = {Louis Shapiro and Atj)\an I}, title = {Proposition}, year = {} }

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Abstract

In this note we call attention to the curious fact that the Fibonacci numbers arise when we look at that familiar example from group theory, the n X n nonsingular upper triangular matrices. Once incidence subgroups are defined the result follows quite easily. Let K be any field with more than two elements and let K * denote the nonzero elements of K. We define Tn to be the group of all nonsingular n x n upper triangular matrices over K. That is Tn = i (a-tj)\ajj = 0 i f /> / , an^K*, a,-j e K\. The key definition is as follows. Definition. A subgroup, H, of Tn is an incidence subgroup if (a) The relations defining H can be given entirely by specifying the domain for each a,f. (b) The two possibilities for each a,;- are an = 1 or a,,- e F*. (c) The two possibilities for a,f when / < / are <?; / = 0 or a,f e F Since He Tn we automatically have a,f = 0 whenever />j. By way of example we have is an incidence subgroup of Td. is a subgroup but not an incidence subgroup since the (1,2) and (1_„3) entries are dependent. 0 / A] | a,h e K \0 0 1 is not a subgroup. We let G ' denote the commutator subgroup of G. Then it is easily shown that For instance which is an incidence subgroup. Our result is the following: