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Let G and A be groups where A acts on G by automorphisms. We say "the action of  A on  G is good" if the equality H = [ H , B ] C H (B) holds for any subgroup B of A and for any B -invariant subgroup H of G. It is straightforward that every coprime action is a good action. In this work, we extend some results due to Ward, Gross, Shumyatsky, Jabara and Meng and Guo under coprime action to good action. [ABSTRACT FROM AUTHOR]

We prove that a group with spectrum {1, 2, 3, 4, 5, 8} is locally finite and is isomorphic to M10, where M10 = A6 ⋅ 2 is the 3-transitive Mathieu group of degree 10, i.e. the corresponding maximal subgroup of the sporadic simple group M11. [ABSTRACT FROM AUTHOR]