Constructing Subgroups of Semi-direct Products via Generalized Derivations

  • Jill Dietz St. Olaf College
  • Akina Khan St. Olaf College
  • Asa Giannini St. Olaf College
  • Michael Schroeder St. Olaf College


There is a well-known correspondence due to Goursat between subgroups of a direct product of groups, A × B, and triples of the form (A1/A2, B1/B2, σ) where A2 ⊲ A1 ≤ A, B2 ⊲ B1 ≤ B, and σ : A1/A2 → B1/B2 is an isomorphism. By contrast, there is a little known correspondence due to Usenko between subgroups of a semi-direct product U ⋊φ H, where φ : H → AutU, and triples of the form (L, R, θ), where L ≤ U, R ≤ H, and θ : R → U is a kind of generalized derivation (crossed homomorphism). While Goursat’s theorem has been used many times to investigate subgroups of direct products, Usenko’s theorem has not, perhaps due to the computational complexity of finding the generalized derivations. In our paper, we find ways of reducing the computational complexity, and show how to use Usenko’s correspondence to determine all of the subgroups of a certain metacyclic p-group.


[1] E. Goursat, Sur les substitutions orthogonales et les divisions regulieres de lespace, Ann. Sci. Ecole Norm. Sup., 6
(1889) 9-102.

[2] H. Heineken, review of “Die Untergruppen von halbdirekten Produkten” by K. Rosenbaum, Mathematical Reviews,
MR0991728 (90c:20032).

[3] K. Rosenbaum, Die Untergruppen von halbdirekten Produkten, Rostock. Math. Kolloq., 35 (1988) 21-30.
[4] V.M. Usenko, Subgroups of semi direct products, Ukranian Math. J., 43 (1991) 982-988.
How to Cite
DIETZ, Jill et al. Constructing Subgroups of Semi-direct Products via Generalized Derivations. Minnesota Journal of Undergraduate Mathematics, [S.l.], v. 4, n. 1, june 2018. ISSN 2378-5810. Available at: <>. Date accessed: 14 july 2020.