Subgroup-cofactorial automorphism-invariant implies left-transitively 2-subnormal

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This article gives the statement and possibly, proof, of an implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., subgroup-cofactorial automorphism-invariant subgroup) must also satisfy the second subgroup property (i.e., left-transitively 2-subnormal subgroup)
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Get more facts about subgroup-cofactorial automorphism-invariant subgroup|Get more facts about left-transitively 2-subnormal subgroup
This article describes a computation relating the result of the Composition operator (?) on two known subgroup properties (i.e., Subgroup-cofactorial automorphism-invariant subgroup (?) and 2-subnormal subgroup (?)), to another known subgroup property (i.e., 2-subnormal subgroup (?))
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Statement

Suppose H \le K \le G are groups such that H is a subgroup-cofactorial automorphism-invariant subgroup of K and K is a 2-subnormal subgroup of G. Then, H is also a 2-subnormal subgroup of G.