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

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.


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
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