Stellenbosch University
Revisiting Isbell’s formulation of the Zassenhaus
lemma
Kishan Dayaram, University of
Johannesburg
SAMS Subject Classification Number: 4, 15, 18
Isbell’s paper [1] shows that the construction in the proof of the Jordan-Hölder-Schreier theorem by Zassenhaus [4] is stronger than what is stated. Moreover Isbell’s formulation shows that the Jordan-Hölder-Schreier theorem is a consequence of the Zassenhaus lemma. We shall show that Isbell’s formulation and proofs of the Zassenhaus lemma and the Jordan-Hölder-Schreier theorem may be extended to the context of the noetherian form. This context, introduced in [3] and further developed in [2], is a self-dual context which covers all group-like structures.
References
[1] J. Isbell, Zassenhaus’ theorem supercedes the Jordan-Hölder theorem, Advances in Mathematics 31 (1979), 101–103.
[2] A. Goswami and Z. Janelidze, Duality in non-abelian algebra IV. Duality for groups and a universal isomorphism theorem, Advances in Mathematics 349 (2019), 781–812.
[3] F. K. van Niekerk, Biproducts and commutators for noetherian forms, Theory and Applications of Categories 34 (2019), 961–992.
[4] H. Zassenhaus, Zum Satz von Jordan-Hölder-Schreier, Abh. Math. Sem. Univ. Hamburg 10 (1934), 106–-108.
