D8 is not potentially fully invariant in D16

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This article gives the statement, and possibly proof, of a particular subgroup or type of subgroup (namely, Dihedral group:D8 (?)) not satisfying a particular subgroup property (namely, Potentially fully invariant subgroup (?)) in a particular group or type of group (namely, Dihedral group:D16 (?)).

Statement

Suppose G is the dihedral group of order 16:

G := \langle a,x \mid a^8 = x^2 = e, xax = a^{-1} \rangle

and H is a subgroup isomorphic to the dihedral group of order 8:

H := \langle a^2, x \rangle

Then, H is not a potentially fully invariant subgroup of G.

Facts used

  1. Fully normalized and potentially fully invariant implies centralizer-annihilating endomorphism-invariant