Minnesota Journal of Undergraduate Mathematics

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

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Abstract

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.