AN EXPLICIT DIAGONAL RESOLUTION FOR A NON-ABELIAN METACYCLIC GROUP. Issue 2 (23rd March 2017)

• Record Type: Journal Article
• Authors: Remez, J. J.
• Is Part Of: Mathematika. Volume 63:Issue 2(2017)
• Page Start: 499
• Page End: 517
• Publication Date: 2017-03-23
• Subjects: 16E05, -- 20C10 (primary); Mathematics -- Periodicals; 510.5
• DOI: 10.1112/S0025579317000018
• Languages: English
• ISSNs: 0025-5793

Abstract

Abstract : We consider the notion of a free resolution. In general, a free resolution can be of any length depending on the group ring under investigation. The metacyclic groups $G(pq)$ however admit periodic resolutions. In the particular case of $G(21)$ it is possible to achieve a fully diagonalized resolution . In order to achieve a diagonal resolution, we obtain a complete list of indecomposable modules over $\unicode[STIX]{x1D6EC}$ . Such a list aids the decomposition of the augmentation ideal (the first syzygy ) into a direct sum of indecomposable modules. Therefore, we are able to achieve a diagonalized map here. From this point it is possible to decompose all of the remaining syzygies in terms of indecomposable modules, leaving a diagonal resolution.