metabelianadj.
/ˌmɛtəˈbiːlɪən/, U.S. /ˌmɛdəˈbiliən/ Mathematics.
1. Of a group: having the property that its factor group by its centre is abelian; having an abelian group of inner automorphisms. Now rare.Quot. 1902 is from a paper first presented at a conference in 1899.
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1900 W. B. Fite in Proc. 49th Meeting Amer. Assoc. Advancem. Sci. 41 If the commutators of a metabelian group G of order pm, p being a prime, form a cyclical group pa, then the number of independent generators of any given order of the group of cogredient isomorphisms G1 must be even.
1902 W. B. Fite in Trans. Amer. Math. Soc. 3 331 We define a Metabelian Group as a group whose group of cogredient isomorphisms is abelian... Hence we could define a metabelian group as a group whose commutators are invariant.
1934 Amer. Jrnl. Math. 56 490 (title) On the metabelian groups which contain a given group H as a maximal invariant Abelian subgroup.
2. Of a group or a Lie algebra: expressible as an extension of one abelian group or Lie algebra by another abelian group or Lie algebra; having the property that its commutator subgroup or subalgebra is abelian.
ΚΠ
1910 Q. Jrnl. Math. 41 224 If G is metabelian, so that its derived group is abelian, the operations (a, b) are all permutable with each other.
1957 Amer. Jrnl. Math. 79 885 If a connected unimodular Lie group has a left invariant Kaehlerian structure, then the group is meta-abelian.
2007 K. H. Hofmann & S. A. Morris Lie Theory Connected Pro-Lie Groups vii. 328 Recall that a Lie algebra is called metabelian, if its commutator algebra is abelian.
This is a new entry (OED Third Edition, December 2001; most recently modified version published online March 2022).
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adj.1900