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AN EXPLICIT DIAGONAL RESOLUTION FOR A NON-ABELIAN METACYCLIC GROUP

Part of: Homological methods Representation theory of groups

Published online by Cambridge University Press:  23 March 2017

J. J. Remez
J. J. Remez*
Affiliation:
Department of Mathematics, University College London, Gower Street, London, WC1E 6BT, U.K. email j.remez@ucl.ac.uk
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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.

MSC classification

Primary: 16E05: Syzygies, resolutions, complexes 20C10: Integral representations of finite groups
Type
Research Article
Information
Mathematika , Volume 63 , Issue 2 , 2017 , pp. 499 - 517
Copyright
Copyright © University College London 2017 

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