Rational product group under the operation (a,b)*(c,d) =(ac,ad+b)
Source:
November 28, 2019
group theoryabstract algebra
Problem Statement
Define the operation (a,b)∘(c,d)=(ac,ad+b).a) Prove that (Q∖{0}×Q,∘) is a group.
b) Let H be an infinite subgroup of (Q∖{0}×Q,∘) that is cyclic and doesn't contain any element of the form (1,q), where q is a nonzero rational. Show that there exist two rational numbers a,b such that
H={(an,b⋅1−a1−an)n∈Z}