I hope to describe how Galois fields can be used to construct: first Camina p-groups P (with [P, x] = P′ for every element x ∈ P − P′) [1]; then semidirect products of an extraspecial normal subgroup by a cyclic group [2]; and finally (in response to a question asked by D. MacHale, and prompted by an example due to R. Heffernan) finite soluble groups G with |Aut G| < |G|. \[x^2\]
References
[1] R. Dark and C.M. Scoppola, On Camina groups of prime power order, J. Algebra 181, pp. 787–802, 1996.
[2] R. Dark, A.D. Feldman and M.D. Pérez-Ramos, Extraspecially irreducible groups, Adv. Group Theory and Appl. 2, pp. 31–65, 2016
