Korean J. Math.  Vol 19, No 2 (2011)  pp.
DOI: https://doi.org/10.11568/kjm.2011.19.2.

### SUBGROUP ACTIONS AND SOME APPLICATIONS

Juncheol Han, Sangwon Park

#### Abstract

Let G be a group and X be a nonempty set and H be a subgroup of G. For a given φ_G : G×X → X, a group action of G on X , we define φ_H :H×X→X, a subgroup action of H on X, by φ_H(h,x) = φ_G(h,x) for all (h,x) ∈ H×X. In this paper, by considering a subgroup action of H on X, we have some results as follows: (1) If H, K are two normal subgroups of G such that H ⊆ K ⊆ G, then for any x ∈ X (orb_{φG} (x) : orb_{φH} (x)) = (orb_{φG} (x) : orb_{φK} (x)) = (orb_{φK} (x) : orb_{φH} (x)); additionally, in case of K ∩ stab_{φG} (x) = {1}, if (orb_{φG} (x) : orb_{φK} (x)) and (orb_{φK} (x) : orb_{φH} (x)) are both finite, then (orb_{φG} (x) : orb{φH} (x)) is finite; (2) If H is a cyclic subgroup of G and stab_{φH} (x) = {1} for some x ∈ X, then orb_{φH} (x) is finite.

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