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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.
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참고문헌 (6)
참고문헌 신청
D. S. Passman / 1977 / The algebraic structure of group rings / John Wiley and Sons, Inc
J. A. Cohen / 1989 / Half-transitive group actions in a compact ring / J. Pure Appl. Algebra 60 : 139 ~ 153
J. Han / 1997 / Regular action in a ring with a nite number of orbits / Comm. Algebra 25 (7) : 2227 ~ 2236
J. Han / 1999 / Group actions in a unit-regular ring / Comm. Algebra 27 (7) : 3353 ~ 3361
Juncheol Han / 2006 / General Linear Group over a Ring of Integers of Modulo k / Kyungpook Mathematical Journal 46 (2) : 255 ~ 260
J. A. Cohen / 1989 / Half-transitive group actions in a compact ring / J. Pure Appl. Algebra 60 : 139 ~ 153
J. Han / 1997 / Regular action in a ring with a nite number of orbits / Comm. Algebra 25 (7) : 2227 ~ 2236
J. Han / 1999 / Group actions in a unit-regular ring / Comm. Algebra 27 (7) : 3353 ~ 3361
Juncheol Han / 2006 / General Linear Group over a Ring of Integers of Modulo k / Kyungpook Mathematical Journal 46 (2) : 255 ~ 260
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