From Baumslag’s “Topics in Combinatorial Group Theory,” chapter V, Exercises 2(3)(iv):

is of order e > 2 if, and only if, for some primitive e-th root of unity .

There’s really nothing to this statement — just put the matrix in Jordan Canonical Form and draw the obvious conclusion — but it really surprised me when I saw it used in an argument. Another way to put this would be that, for a 2×2 matrix, the trace and determinant determine the eigenvalues (which is equally obvious).

(The problem in the cases *e = 1* and *e = 2* is that then the eigenvalues are identical, which means that the matrix isn’t necessarily diagonal when it’s in Jordan Canonical Form — it could be a 2×2 Jordan block.)

From this and one other fact it follows that the elements *a*, *b*, and *ab* in the group actually have the desired orders.

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