More on regular subgroups of the affine group. (English) Zbl 1360.20044

Pellegrini, M. A.; Tamburini Bellani, M. C.

Linear Algebra Appl. 505, 126-151 (2016).

Summary: This paper is a new contribution to the study of regular subgroups of the affine group \(\operatorname{AGL}_n(\mathbb{F})\), for any field \(\mathbb{F}\). In particular we associate to each partition \(\lambda \neq(1^{n + 1})\) of \(n + 1\) abelian regular subgroups in such a way that different partitions define non-conjugate subgroups. Moreover, we classify the regular subgroups of certain natural types for \(n \leq 4\). Our classification is equivalent to the classification of split local algebras of dimension \(n + 1\) over \(\mathbb{F}\). Our methods, based on classical results of linear algebra, are computer free.

MSC:

20G15	Linear algebraic groups over arbitrary fields
15A21	Canonical forms, reductions, classification
16L30	Noncommutative local and semilocal rings, perfect rings
20B35	Subgroups of symmetric groups

Keywords:

regular subgroup; local algebra; Jordan canonical form

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