Characteristic of CDIN implies CDIN

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This article describes a computation relating the result of the Composition operator (?) on two known subgroup properties (i.e., Characteristic subgroup (?) and CDIN-subgroup (?)), to another known subgroup property (i.e., CDIN-subgroup (?))
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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., characteristic subgroup) must also satisfy the second subgroup property (i.e., left-transitively CDIN-subgroup)
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Statement

Statement with symbols

Suppose H \le K \le G are groups such that H is characteristic in K and K is a CDIN-subgroup of G. Then H is a CDIN-subgroup of G.

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