# A Beckman-Quarles type theorem for finite desarguesian by Benz W.

By Benz W.

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Q1 we obtain P m g − f = Qm (g − f ) . We shall now prove that Qm g −→ Q0 g for each g; since P0 g = Q0 (g − f ) + f this would prove the result. But the statement about Qm comes from the following general result about abstract Hilbert space. 11. Let H be a Hilbert space and Qi the projection of H onto a subspace Ni ⊂ H(1 ≤ i ≤ k). Let N0 = ∩k1 Ni and Q0 : H −→ N0 the projection. Then if Q = Qk . . Q1 Qm g −→ Q0 g for each g ∈ H, . Since Q is a contraction ( Q ≤ 1) we begin by proving a simple lemma about such operators.

Let g, γ ∈ G, x ∈ gK, ξ = γH. Then ˇx is an orbit of K g , ξ is an orbit of H γ , and ˇx = K g /Lg , ξ = H γ /Lγ . Proof. By definition (3) ˇx = {δH : δH ∩ gK = ∅} = {gkH : k ∈ K} which is the orbit of the point gH under gKg −1 . The subgroup fixing gH is gKg −1 ∩ gHg −1 = Lg . Also (3) implies ˇx = g · ˇx0 ξ = γ · ξ0 , where the dot · denotes the action of G on X and Ξ. 3. Consider the subgroups KH HK = {k ∈ K : kH ∪ k −1 H ⊂ HK} = {h ∈ H : hK ∪ h−1 K ⊂ KH} . The following properties are equivalent: (a) K ∩ H = K H = HK .

Q1 we obtain P m g − f = Qm (g − f ) . We shall now prove that Qm g −→ Q0 g for each g; since P0 g = Q0 (g − f ) + f this would prove the result. But the statement about Qm comes from the following general result about abstract Hilbert space. 11. Let H be a Hilbert space and Qi the projection of H onto a subspace Ni ⊂ H(1 ≤ i ≤ k). Let N0 = ∩k1 Ni and Q0 : H −→ N0 the projection. Then if Q = Qk . . Q1 Qm g −→ Q0 g for each g ∈ H, . Since Q is a contraction ( Q ≤ 1) we begin by proving a simple lemma about such operators.