Characteristic not implies fully invariant in class three maximal class p-group

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Statement

Let p be any prime number. There exists a p-group that is of class three that is a Maximal class group (?) with a Characteristic subgroup (?) that is not fully invariant.

Related facts

Similar facts

Opposite facts

Proof

For odd primes

Suppose p is an odd prime. Let K be the wreath product of groups of order p and G be K/[[[K,K],K],K]. G is thus a semidirect product of an elementary abelian group of order p^3 and a cyclic group of order p. Note that for p > 3, G has exponent p but for p = 3, G has exponent 9.

Consider H = C_G([G,G]), the centralizer of commutator subgroup. H is a characteristic subgroup. Consider now an element x of order p outside H (such an element exists because of the way the group is defined as a semidirect product) and a subgroup L of G of index p other than H, that does not contain x. Consider the retraction from G to \langle x \rangle with kernel L. This retraction doesn't preserve H, so H is not fully invariant.

For the prime two

Further information: dihedral group:D16

Consider the dihedral group:

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

Let H = \langle a \rangle. Then, H = C_G([G,G]), so H is characteristic in G. Consider now the subgroup L = \langle a^2, x \rangle and the element ax. Consider the retraction to the subgroup \langle ax \rangle with kernel L. H is not preserved under this retraction (for instance, a gets mapped to x), so it is not fully invariant.