By Ilijas Farah

This ebook is meant for graduate scholars and examine mathematicians attracted to set idea.

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Extra info for Analytic Quotients: Theory of Liftings for Quotients over Analytic Ideals on the Integers

Example text

Let Γ every element of Γ. Euc(2) be a finite subgroup. Then there is a point of R2 fixed by Proof. Let the distinct elements of Γ be F1 , . . , Fn , where n = |Γ|. Let p ∈ R2 be (the position vector of) any point. Define 1 1 F1 (p) + · · · + Fn (p). n n For any k = 1, . . , n, by a result on Problem Sheet 4, we have 1 1 Fk (p0 ) = Fk F1 (p) + · · · + Fk Fn (p). n n p0 = Now if Fk Fi = Fk Fj , then Fk−1 Fk Fi = Fk−1 Fk Fj and so Fi = Fj . Also, every Fr can be written as Fr = Fk (Fk−1 Fr ) where Fk−1 Fr ∈ Γ has the form Fk−1 Fr = Fs for some s and therefore Fr = Fk Fs .

Is it the only one? (c) Find a suitable similarity transformation ϕ for which ϕ∗ Γ O(2). 14. Let Γ Euc(2) be a subgroup containing the isometries F, G : R2 −→ R2 . (a) If F and G are reflections in two distinct parallel lines, show that there is no point fixed by all the elements of Γ. Deduce that Γ is infinite. (b) If F is the reflection in a line L and G is a non-trivial rotation about a point p not on L, show that Γ is infinite. 15. Let Γ Euc(2) be a subgroup containing the isometries F, G : R2 −→ R2 and suppose that these generate Γ in the sense that every element of Γ is obtained by repeatedly composing powers of F and G.

Let Γ every element of Γ. Euc(3) be a finite subgroup. Then there is a point of R2 fixed by There are obvious notions of similarity generalizing those for Euc(2) to Euc(3). We define the 3 × 3 orthogonal group to be O(3) = {(A | 0) : AT A = I3 } Euc(3) and the 3 × 3 special orthogonal group to be SO(3) = {(A | 0) : AT A = I3 , det A = 1} O(3) Euc(3). , Euc(2) Euc(3), by thinking of Euc(2) as consisting of isometries of R3 that fix all the points on the z-axis. Thus    a11 a12 0   Euc(2) = (A | (t1 , t2 , 0)) : A = a21 a22 0 Euc(3).

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