  In this note we introduce the notion of a transcendental group, that is, a
subgroup $G$ of the topological group $\mathbb{C}$ of all complex numbers such
that every element of $G$ except $ 0$ is a transcendental number. All such
topological groups are separable metrizable zero-dimensional torsion-free
abelian groups. Further, each transcendental group is homeomorphic to a
subspace of $\mathbb{N}^{\aleph_0}$, where $\mathbb{N}$ denotes the discrete
space of natural numbers. It is shown that (i) each countably infinite
transcendental group is a member of one of three classes, where each class has
$\mathfrak{c}$ (the cardinality of the continuum) members -- the first class
consists of those isomorphic as a topological group to the discrete group $\ZZ$
of integers, the second class consists of those isomorphic as a topological
group to $\ZZ\times \ZZ$, and the third class consists of those homeomorphic to
the topological space $\QQ$ of all rational numbers; (ii) for each cardinal
number $\aleph$ with $\aleph_0< \aleph\le \cc$, there exist $2^\aleph$
transcendental groups of cardinality $\aleph$ such that no two of the
transcendental groups are isomorphic as topological groups or even
homeomorphic; (iii) there exist $\mathfrak{c}$ countably infinite
transcendental groups each of which is homeomorphic to $\QQ$ and algebraically
isomorphic to a vector space over the field $\AAA$ of all algebraic numbers
(and hence also over $\QQ$) of countably infinite dimension; (iv) $\RR$ has
$2^\cc$ transcendental subgroups, each being a zero-dimensional metrizable
torsion-free abelian group, such that no two of the transcendental groups are
isomorphic as topological groups or even homeomorphic.
