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Here's a comment that might help somebody else: if A(x) = a<sub>1 x + a<sub>2 x^2 + ... is a generating function, then Q is the linear transformation that sends (a<sub>1, a<sub>2, ...) to (A(q), A(q^2), ...).
For what it's worth, I never had an "algebra sucks" phase because for me, at least in high school, the big motivation for algebra came from number theory, and I've always been interested in number theory.
That argument's a standard trick: if you start with an inner product (u, v), define a new one by (u, v)' = 1/|G| sum (gu, gv). This inner product is G-invariant so the representation is orthogonal with respect to it.
As well as a nice presentation, a good notion of cycle decomposition, and so forth. The set of all bijections from a countable set to itself, on the other hand, is terrible; one can imagine a bijection which, for example, encodes Chaitin's constant.