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Let $𝔤$ be a simple Lie algebra and ${𝔄𝔟}^{o}$ the poset of non-trivial abelian ideals of a fixed Borel subalgebra of $𝔤$ . In [8], we constructed a partition ${𝔄𝔟}^{o} = ⊔_{μ} {𝔄𝔟}_{μ}$ parameterised by the long positive roots of $𝔤$ and studied the subposets ${𝔄𝔟}_{μ}$ . In this note, we show that this partition is compatible with intersections, relate it to the Kostant-Peterson parameterisation and to the centralisers of abelian ideals. We also prove that the poset of positive roots of $𝔤$ is a join-semilattice.

Abelian Normal Subgroups of M-Groups.

I. Martin Isaacs (1983)

Mathematische Zeitschrift

Abelian p-adic Group Rings.

Eugene Spiegel (1975)

Commentarii mathematici Helvetici

Abelian pro-countable groups and orbit equivalence relations

Maciej Malicki (2016)

Fundamenta Mathematicae

We study a class of abelian groups that can be defined as Polish pro-countable groups, as non-archimedean groups with a compatible two-sided invariant metric or as quasi-countable groups, i.e., closed subdirect products of countable discrete groups, endowed with the product topology. We show that for every non-locally compact, abelian quasi-countable group G there exists a closed L ≤ G and a closed, non-locally compact K ≤ G/L which is a direct product of discrete countable groups....

Abelian profinite groups

Krzysztof Krupiński (2005)

Fundamenta Mathematicae

Abelian quasinormal subgroups of groups

Stewart E. Stonehewer, Giovanni Zacher (2004)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Let $G$ be any group and let $A$ be an abelian quasinormal subgroup of $G$ . If $n$ is any positive integer, either odd or divisible by $4$ , then we prove that the subgroup $A^{n}$ is also quasinormal in $G$ .