Normal not implies finite-pi-potentially characteristic in finite
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., normal subgroup) need not satisfy the second subgroup property (i.e., finite-pi-potentially characteristic subgroup)
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- Every prime divisor of the order of is also a prime divisor of the order of .
- is a characteristic subgroup of .
- Every finite p-group is a subgroup of a finite p-group that is not characteristic in any finite p-group properly containing it
By fact (1), we can definitely find a finite -group with the property that is not a characteristic subgroup in any finite -group properly containing it.
Let be the direct product of and a cyclic group of order . is thus a normal subgroup of . However, for any finite -group containing , is a proper subgroup of it and therefore not a characteristic subgroup of it. Hence, cannot be made to be characteristic in any finite -group that contains .