Characteristic not implies normal-isomorph-free

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This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties, when the big group is a finite group. That is, it states that in a finite group, every subgroup satisfying the first subgroup property (i.e., characteristic subgroup) need not satisfy the second subgroup property (i.e., normal-isomorph-free subgroup)
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It is possible to have a finite group G and subgroups H,K of G such that H is a characteristic subgroup of G, K is a normal subgroup of G, and H,K are isomorphic groups.

Facts used

  1. Characteristic not implies characteristic-isomorph-free in finite (combined with normal-isomorph-free implies characteristic-isomorph-free)
  2. Characteristic-isomorph-free not implies normal-isomorph-free in finite (combined with characteristic-isomorph-free implies characteristic)
  3. Characteristic not implies series-isomorph-free (combined with normal-isomorph-free implies series-isomorph-free)


The proof could be obtained using an example for any of the stronger facts (1)-(3).