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Rings and covered groups. II. (English) Zbl 1260.16025

This paper, concerning covers of groups and the rings of functions determined by these covers, resolves some open questions from part I [G. A. Cannon, C. J. Maxson and K. M. Neuerburg, J. Algebra 320, No. 4, 1586-1598 (2008; Zbl 1176.16035)], and also adds some new results.
A collection \(\mathcal C=\{C_\alpha:\alpha\in\Sigma\}\) of subgroups of a group \((G,+)\) (not necessarily Abelian) is a ‘cover’ of \(G\) if \(G=\bigcup_{\alpha\in\Sigma}C_\alpha\). In this paper, only covers comprising Abelian subgroups (such as the cyclic subgroups) of \(G\) are considered. The ‘ring determined by the cover \(\mathcal C\)’, denoted by \(\mathcal R(\mathcal C)\), is defined as the set of all functions \(f\colon G\to G\) with the property that \(f|_{C_\alpha}\in\text{End}(C_\alpha)\) for all \(\alpha\in\Sigma\), endowed with the operations of pointwise addition and function composition.
A relation called ‘order-connectedness’ on \(G\) (with respect to a cover \(\mathcal C\)) is defined and this concept is used to describe exactly (in case \(G\) is finite nonabelian and \(\mathcal C\) consists of all Abelian subgroups of \(G\)) when the ring \(\mathcal R(\mathcal C)\) is generated by the identity map. The case where \(G\) is Abelian was already solved [in loc. cit.].
Further results concern the simplicity of \(\mathcal R(\mathcal C)\). For example, if \(|G|=p^n\), (\(p\) prime and \(n\leq 5\)), \(\exp(G)=p\) and \(G\notin\mathcal C\), then \(\mathcal R(\mathcal C)\) is simple if and only if \(\mathcal R(\mathcal C)\cong\mathbb Z_p\); or, if \(G\) is a finite \(p\)-group of exponent \(p\), and \(\mathcal C\) is a finite cover of \(G\), then \(\mathcal R(\mathcal C)\) is simple only if \(\bigcap\mathcal C=\{0\}\) or \(\mathcal C=\{G\}\). It is also proved that any field is isomorphic to \(\mathcal R(\mathcal C)\) for suitable \(G\) and \(\mathcal C\).
The paper concludes with a section where it is assumed that all subgroups in \(\mathcal C\) have order \(p^2\), \(p\) prime. One of the main results here states that, in case \(G\) is a finite \(p\)-group of exponent \(p\) (and all subgroups in \(\mathcal C\) are of order \(p^2\)), then \(\mathcal R(\mathcal C)\) is simple if and only if it is isomorphic to \(\mathbb Z_p\).

MSC:

16S50 Endomorphism rings; matrix rings
16D60 Simple and semisimple modules, primitive rings and ideals in associative algebras
16Y30 Near-rings
20D45 Automorphisms of abstract finite groups
20D30 Series and lattices of subgroups
20D15 Finite nilpotent groups, \(p\)-groups

Citations:

Zbl 1176.16035
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References:

[1] Cannon G. A., Journal of Algebra 320 pp 1586– (2008) · Zbl 1176.16035 · doi:10.1016/j.jalgebra.2008.05.008
[2] Karzel H., Results in Mathematics 9 pp 70– (1986)
[3] McDonald B. R., Finite Rings with Identity (1974) · Zbl 0294.16012
[4] Mason G., Nearrings and Nearfields: Proceedings of a conference held at Mathematische Forschungsinstitut Oberwolfach (1989)
[5] Maxson C. J., Algebra and Discrete Mathematics 1 pp 59– (2009)
[6] Maxson C. J., Results in Mathematics 16 pp 140– (1989)
[7] Maxson C. J., Journal of Algebra 315 (2) pp 541– (2007) · Zbl 1131.16022 · doi:10.1016/j.jalgebra.2007.02.047
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