Let n be given. Can you find nxn invertible matrices T and S over Z

_{2}such that T

^{2}S = ST? If so, can you furthermore have both S and T be 'primitive' in the sense that neither S

^{k}nor T

^{k}are the identity for any positive k < 2

^{n}-1?

Is this possible for some n, for no n, for most n, ...?

An example of the first condition being satisfied in a continuous setting --- affine transformations over real vectors --- is T being a translation and S being a scaling by a factor of two: translate and scale up is the same thing as scale up and translate twice as much.

The second condition is just asking that S and T considered separately are good LFR pseudorandom number generators.

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Hmm, my gut instinct after thinking about it for five minutes is: probably true for hardly any n, if any at all.

I'm demanding that all the 2

^{n}-1 powers of T are distinct, and that (T, T^2, T^4, T^8,...) are all in the same conjugacy class, and also that (T^3, T^6, T^12, T^24, ...) are all in the same conjugacy class, and also (T^5, T^10, T^20, T^40, ...), and so on starting at every odd power of T. So I'm making really strong constraints on the structure of the conjugacy classes of GF(Z

_{2}), and I bet reality doesn't conform to them.