Subgroup actions and some applications
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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.
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.
참고문헌 (6)
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