By Heinonen J., Kilpelftine T.

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**Example text**

1. If H is a group such that G ⊕ G ∼ = G ⊕ H then G ∼ = H. 2. If H is locally isomorphic to G then there is a group K unique to H such that G⊕G ∼ = H ⊕ K. Proof: 1. Given G ⊕ G ∼ = G ⊕ H then by definition of λ2 , (AG (G)) = (E(G))(E(G)) = λ2 ((G ⊕ G)) = λ2 ((G ⊕ H )) = (E(G))(AG (H )) = (AG (H )). 2, G ∼ = H. 18 Class number of an abelian group 2. Suppose that H is locally isomorphic to G. Then (H ) ∈ (G) so that (AG (H )) ∈ Pic(E(G)). 15 there is a unique group K that is locally isomorphic to G such that (AG (K)) = λ1 ((K)) = (AG (H ))−1 .

Fuchs. 7. 10] Let A be an rtffr group and let n = 0 ∈ Z be such that nA = A. Assume that there is a finite rank torsion-free group G such that 1. End(G) = Z, 2. Hom(A, G) = Hom(G, A) = 0, and 3. There is an epimorphism β : G → A/nA. If A has the cancellation property then every unit of End(A)/nEnd(A) lifts to a unit of End(A). 8. 1. If each unit of E/τ lifts to a unit of E then h(G) = h(G). Proof: Suppose that units of E/τ lift to units of E. 1) is a surjection, so that Pic(E) ∼ = Pic(E). 2, h(G) = card(Pic(E)) = card(Pic(E)) = h(G).

Proof: Suppose that G n has a -unique decomposition. Let m > 0 be an integer and let G m = H1 ⊕ · · · ⊕ Ht for some integer t > 0 and some indecomposable groups H1 , . . , Ht . 2 each Hi is locally isomorphic to G, so by Jónsson’s theorem, m = t. Then (G n )m = H1n ⊕ · · · ⊕ Hmn so by hypothesis Hi ∼ = G for each 22 Class number of an abelian group i = 1, . . , t. Thus G has a proof is complete. -unique decomposition. 35. Let G be a cocommutative strongly indecomposable rtffr group. Then G has a -unique decomposition iff G ⊕ G has a unique decomposition.

### A -superharmonic functions and supersolutions of degenerate elliptic equations by Heinonen J., Kilpelftine T.

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