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- W2014298467 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_{ φ G} ( x ) : orb_{ φ H} ( x )) = ( orb_{ φ G} ( x ) : orb_{ φ K} ( x )) = ( orb_{ φ K} ( x ) : orb_{ φ H} ( x )); additionally, in case of K ∩ stab_{ φ G} ( x ) = { 1 } , if ( orb_{ φ G} ( x ) : orb_{ φ K} ( x )) and ( orb_{ φ K} ( x ) : orb_{ φ H} ( x )) are both finite, then ( orb_{ φ G} ( x ) : orb{ φ H} ( x )) is finite; (2) If H is a cyclic subgroup of G and stab_{ φ H} ( x ) = { 1 } for some x ∈ X , then orb_{ φ H} ( x ) is finite." @default.
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- W2014298467 date "2011-06-30" @default.
- W2014298467 modified "2023-09-23" @default.
- W2014298467 title "SUBGROUP ACTIONS AND SOME APPLICATIONS" @default.
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