By Wu X.

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Proposition d exp0 = id. There is an open neighbourhood U of 0 in g and an open neighbourhood V of e in G such that exp: U → V is a C ∞ diffeomorphism. Then the inverse map exp−1 : V → U is denoted by log. Proof The second statement follows by the inverse function theorem. For the proof of the first statement let A ∈ g, α(t) := tA and β(t) := exp(tA). Thus α represents A ∈ T0 g and A = β ′ (0) = d exp0 (A). Hence d exp0 = id. 18 Proposition Let G be a Lie group and put g := Te G. 3) as |A|, |B| → 0 in g.

If already |t| < ε then by the local homomorphism property of α the new value of α(t) is the same as the old value. 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 .

F ) . )))(x) k 1 ! . km ! + O(|t|n+1 ) as t → 0. 22 have Proposition Let G and g be as above. f ). Proof Let x ∈ G. We will expand f (x exp(tA) exp(tB) exp(−tA) exp(−tB)) in two different ways as a Taylor series in t up to degree 2, where t → 0 in R. Then we obtain the result by equality of second degree terms in both expansions. f ))(x) + O(|t|3 ). f )(x) + O(|t|3 ). 23 Corollary Let G be a Lie group with g := Te G and Lie(G) the Lie algebra of left invariant vector fields on G. Put [A, B] := b(A, B) (A, B ∈ g).

### 8-ranks of Class Groups of Some Imaginary Quadratic Number Fields by Wu X.

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