Abstract.Let T be a subgroup of finite index in SLn(Z) with n > 3. We show that every continuous action of T on the circle 51 or on the real line R factors through an action of a finite quotient of T. This follows from the algebraic fact that central extensions of T are not right orderable. (In particular, T is not right orderable.) More generally, the same results hold if T is any arithmetic subgroup of any simple algebraic group G over Q , with Q-rank(G) > 2.

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We explicitly determine all of the transitive groups of degree p 2 , p a prime, whose Sylow p-subgroup is not isomorphic to the wreath product Zp ≀ Zp. Furthermore, we provide a general description of the transitive groups of degree p 2 whose Sylow p-subgroup is isomorphic to Zp ≀ Zp, and explicitly determine most of them. As applications, we solve the Cayley Isomorphism problem for Cayley objects of an abelian group of order p 2 , explicitly determine the full automorphism group of Cayley graphs of abelian groups of order p 2 , and find all nonnormal Cayley graphs of order p 2 .

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