Metabelian group

In the mathematical field of group theory , Metabelian groups are a class of groups that can be broken down in a certain way as the product of two Abelian groups.

definition

A group is metabelian if all commutators commutate with one another, i.e. if the equation for all ${\displaystyle G}$ ${\displaystyle x,y,z,w\in G}$

${\displaystyle (xyx^{-1}y^{-1})(zwz^{-1}w^{-1})=(zwz^{-1}w^{-1})(xyx^{-1}y^{-1})}$

applies. In other words, let the commutator subgroup be an Abelian group . ${\displaystyle \left[G,G\right]}$

An equivalent condition is that there are Abelian groups and an exact sequence ${\displaystyle A,B}$

${\displaystyle 0\to A\to G\to B\to 0}$

gives. In the English-language literature, Metabelian groups are therefore also referred to as abelian-by-abelian groups .

Examples

• The group of the 2-dimensional regular upper triangular matrices is Metabelian. The commutator subgroup in this case is the Abelian group of triangular matrices of the form , the quotient group is isomorphic to the group of regular diagonal matrices . ${\displaystyle \left({\begin{array}{cc}*&*\\0&*\end{array}}\right)}$ ${\displaystyle \left({\begin{array}{cc}1&*\\0&1\end{array}}\right)}$ ${\displaystyle G/\left[G,G\right]}$
• The group of affine transformations , any body metabelsch. Their commutator group is the Abelian group of translations , the quotient group is isomorphic to the group of homotheties . ${\displaystyle x\mapsto ax+b}$ ${\displaystyle a\neq 0}$ ${\displaystyle x\mapsto x+b}$ ${\displaystyle G/\left[G,G\right]}$ ${\displaystyle x\mapsto ax}$
• The group of orientation-maintaining isometries of the Euclidean plane is Metabelian, their commutator group is the Abelian group of displacements , the quotient group is isomorphic to the rotational group . ${\displaystyle G/\left[G,G\right]}$ ${\displaystyle SO(2)}$
• Abelian groups are Metabelian.
• A nonabelian resolvable group is metabelian if and only if it has a subnormal series of length . ${\displaystyle 2}$
• The symmetric group is metabelian if and only if is. ${\displaystyle S_{n}}$ ${\displaystyle n\leq 3}$
• Every dihedral group and the infinite dihedral group are metabelian. ${\displaystyle D_{n}}$ ${\displaystyle D_{\infty }}$
• The holomorph of a cyclic group is metabelian.
• The Lamplighter group is metabelian.
• Subgroups , quotient groups, and direct products of metabolic groups are again metabolic.

literature

• Specht, Wilhelm: group theory. Springer-Verlag, Berlin-Göttingen-Heidelberg, 1956. vii + 457 pp.
• Meier-Wunderli, H .: Metabel groups. Comment. Math. Helv. 25, (1951). 1-10, doi: 10.1007 / BF02566442 .
• Kaniuth, Eberhard; Thoma, Elmar: Characters of Metabelischer Groups. Arch. Math. (Basel) 20 1969 4–9, doi: 10.1007 / BF01898984 .
• Bertram Huppert : Finite Groups I (=  Basic Teachings of the Mathematical Sciences . No. 134 ). Reprint of the first edition. Springer-Verlag , Berlin 1979, ISBN 3-642-64982-3 , p. 39 .

Web links

• Metabelian Groups (group props)