By Lucchhini A.

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**Example text**

Note that α thus defined on (−kε, kε) is C ∞ . If |s|, |t|, |s + t| < kε then we get α(s + t) = α(s)α(t). Finally, the definition is independent of k since α(t/k)k = α(t/(kl))kl = α(t/l)l if |t/k|, |t/l| < ε. 1) for all t ∈ R. 13). 1) as a system of differential equations on Mn (C ) and, by restriction, also as a system of differential equations on the submanifold G. 14 asssociates in the case of a linear Lie groups G with A ∈ g the C ∞ -homomorphism t → etA . This suggests the definition of the abstract exponential mapping in the case of a general Lie group.

We now put g := Te G. For A ∈ g let ξA : R → G ′ be the unique C ∞ -homomorphism such that ξ A (0) = A. 1), we see that the map (t, A) → ξA (t): R × g → G is C∞ . Also observe that ξA (st) = ξsA (t). Now define the exponential map exp: g → G by exp(A) := ξA (1). Then exp(tA) = ξA (t). So exp: g → G is C ∞ and t → exp(tA): R → G is a C∞ homomorphism with derivative at 0 equal to A. 17 Let G, g be as above. Then d exp0 : T0 g → Te G. But T0 g can be identified with g and Te G = g. Hence d exp0 linearly maps g to g.

Put [A, B] := b(A, B) (A, B ∈ g). f . Ex. 24 Let G be the Lie group defined as the set {(a, b) ∈ R2 | a = 0} with multiplication rule (a, b)(c, d) = (ac, ad + b). Give a basis v, w for the Lie algebra Lie(g) of left invariant vector fields on G and compute the commutator [v, w] = vw − wv as a linear combination of v and w. 8 Ex. , G = R3 with multiplication rule (a, b, c)(a′, b′ , c′ ) = (a + a′ , b + b′ , c + c′ + ab′ ). Give a basis u, v, w for the Lie algebra Lie(g) of left invariant vector fields on G and compute the commutators [u, v], [u, w] and [v, w] as a linear combination of u, v, w.

### A 2-generated just-infinite profinite group which is not positively generated by Lucchhini A.

by Jason

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