In this paper we investigate the action of Polish groups (not necessary abelian) on uncountable spaces. We consider two main situations. First, when orbits given by group are small and second family at most cou

Let K be any field, K(x1, . . . , xn) be the rational function field of n variables over K, and Sn and An be the symmetric group and the alternating group of degree n respectively. For any a ∈ K \ {0}, define an action of Sn on K(x1, . . . , xn) by σ · xi = xσ(i) for σ ∈ An and σ · xi = a/xσ(i) for σ ∈ Sn \ An. Theorem. For any field K and n = 3, 4, 5, the fixed field K(x1, . . . , xn) Sn is ra...

Every action of a group on a set decomposes the set into orbits. The group acts on each of the orbits and an orbit does not have sub-orbits (unequal orbits are disjoint), so the decomposition of a set into orbits could be considered as a “factorization” of the set into “irreducible” pieces for the group action. Our focus here is on these irreducible parts, namely group actions with a single orbit.

We introduce group actions on the integer partitions and their variances. Using generating functions and Burnside’s lemma, we study arithmetic properties of the counting functions arising from group actions. In particular, we find a modulo 4 congruence involving the number of ordinary partitions and the number of partitions into distinct parts.

We give a bound on the reconstructibility of an action G ֌ X in terms of the reconstructibility of a the action N ֌ X, where N is a normal subgroup of G, and the reconstructibility of the quotient G/N . We also show that if the action G ֌ X is locally finite, in the sense that every point is either in an orbit by itself or has finite stabilizer, then the reconstructibility of G ֌ X is at most t...