Baumslag-Solitar group:BS(1,2)

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VIEW FACTS ABOUT THIS GROUP: All factsGroup property satisfactionsSubgroup property satisfactionsGroup property dissatisfactionsSubgroup property satisfactions
VIEW DEFINITIONS USING THIS AS EXAMPLE: All terms |Group properties |Subgroup properties|
VIEW FACTS USING THIS AS EXAMPLE: All facts |Group property non-implications |Subgroup property non-implications |Group metaproperty dissatisfactionsSubgroup metaproperty dissatisfactions

Definition

This group is defined as the Baumslag-Solitar group with parameters 1 and 2, i.e., the group $BS(1,2)$. Explicitly, it is defined by means of the following presentation:

$\langle a,b\mid bab^{-1}=a^{2}\rangle$

Group properties

Property	Satisfied?	Explanation
finitely generated group	Yes	See presentation above
finitely presented group	Yes	See presentation above
one-relator group	Yes	See presentation above
word-hyperbolic group	No
solvable group	Yes	The normal closure of $a$ is isomorphic to the group $\mathbb {Z} [1/2]$, i.e., the group of 2-adic rationals. This is an abelian normal subgroup. The quotient group is isomorphic to $\langle b\rangle$, which is an abelian group isomorphic to the group of integers.
Noetherian group	No	The normal closure of $a$ is isomorphic to the subgroup $\mathbb {Z} [1/2]$, i.e., the group of 2-adic rationals. This is not finitely generated, because any finite generating set has an upper bound on possible denominators.
nilpotent group	No
abelian group	No
polycyclic group	No
supersolvable group	No