1. Introduction 5 We may iterate this argument (for a fixed, but small, value of ε) and conclude that Φ(t0/2k) = exp(B/2k) for all k = 0, 1, 2,... . By the homomorphism property and (1.1) we thus have Φ(qt0) = exp(qB) whenever q is a dyadic rational, i.e., a rational of the form a/2k for some integer a and natural number k. By continuity we thus have Φ(st0) = exp(sB) for all real s. Setting A := B/t0 we conclude that Φ(t) = exp(tA) for all real t, which gives existence of the representation and also real an- alyticity and smoothness. Finally, uniqueness of the representation Φ(t) = exp(tA) follows from the identity A = d dt exp(tA)|t=0. Exercise 1.0.2. Generalise Proposition 1.0.1 by replacing the hypothesis that Φ is continuous with the hypothesis that Φ is Lebesgue measurable. (Hint: Use the Steinhaus theorem, see e.g. [Ta2011, Exercise 1.6.8].) Show that the proposition fails (assuming the axiom of choice) if this hypothesis is omitted entirely. Note how one needs both the group-like structure and the weak reg- ularity in combination in order to ensure the strong regularity neither is suﬃcient on its own. We will see variants of the above basic argument throughout the course. Here, the task of obtaining smooth (or real ana- lytic structure) was relatively easy, because we could borrow the smooth (or real analytic) structure of the domain R and range Mn(C) but, somewhat remarkably, we shall see that one can still build such smooth or analytic structures even when none of the original objects have any such structure to begin with. Now we turn to a second illustration of the above principles, namely Jordan’s theorem [Jo1878], which uses a discreteness hypothesis to upgrade Lie type structure to nilpotent (and in this case, abelian) structure. We shall formulate Jordan’s theorem in a slightly stilted fashion in order to emphasise the adherence to the above-mentioned principles. Theorem 1.0.2 (Jordan’s theorem). Let G be an object with the following properties: (1) (Group-like object) G is a group.

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