Characteristic not implies isomorph-free in finite group

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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., isomorph-free subgroup)
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Statement

Statement with symbols

There exists a finite group G and a characteristic subgroup H of G such that H is not an isomorph-free subgroup of G. In other words, there exists another subgroup K of G that is isomorphic to H.

Related facts

Proof

Example of the dihedral group

Further information: dihedral group:D8, subgroup structure of dihedral group:D8, center of dihedral group:D8

Let G be the dihedral group of order eight, given as follows, where e denotes the identity element of G:

G = \langle a,x \mid a^4 = x^2 = e, xax = a^{-1} \rangle.

Let H be the center of G. H is a subgroup of order two generated by a^2.

  • H is characteristic.
  • H is not isomorph-free: The subgroup \langle x \rangle of G is isomorphic to H.