Every abelian subgroup of a Gromov hyperbolic group is virtually cyclic.

The finitely generated condition is essential here: Q {\displaystyle \mathbb {Q} } is torsion-free but not free abelian.

A finite group is a group whose underlying set is finite.

All Borel subgroups of a given group are conjugate.

It is always cyclic or the product of two cyclic groups.

What do you mean by gangsters and a gangter country ?

The abelian group of translations is a normal subgroup while the Lorentz group is a subgroup, the stabilizer of a point.

Zero is the identity element in an additive group or the additive identity of a ring.

A G-module M is an abelian group M together with a group action of G on M as a group of automorphisms.

The dual group of a locally compact abelian group is used as the underlying space for an abstract version of the Fourier transform.

Generally speaking, negation is an automorphism of any abelian group, but not of a ring or field.

The function F composes output of a plurality of the mapping tables Ti (0≤ i≤ n; n≥1) using an Abelian group operator ⊗.

More generally, any nontrivial group with no proper nontrivial subgroup must be cyclic of prime order.

The dyadic rationals are the direct limit of infinite cyclic subgroups of the rational numbers,

Moreover, any function on a finite group can be recovered from its discrete Fourier transform.