An infinite Abelian
-group
all proper subgroups of which are cyclic (cf.
Cyclic group).
There exists for each prime number
a quasi-cyclic group, and it is unique up to an
isomorphism. This group is isomorphic to the multiplicative
group of all roots of the equations
in the field of complex numbers with the
usual multiplication, and also to the quotient group
,
where
is the additive group of the field of rational
-adic
numbers and
is the additive group of the ring of all
-adic
integers. A quasi-cyclic group is the union of an ascending chain of cyclic groups
of orders
,
;
more precisely, it is the inductive limit
with respect to the inductive system
.
This group can be defined in terms of generators and
relations as the group with countable system of generators
and relations
Quasi-cyclic groups are the only infinite Abelian (and also the only
locally-finite infinite) groups all subgroups of which are finite. The
question of the existence of infinite non-Abelian groups with this
property is still unsolved
(
1978)
and constitutes one of the
problems
of
O.Yu. Shmidt.
Quasi-cyclic groups are divisible Abelian groups (cf.
Divisible group),
and each divisible Abelian group is the direct sum of a set of groups
that are isomorphic to the additive group of rational
numbers and to quasi-cyclic groups for certain prime numbers
.
Groups of type
are maximal
-subgroups
of the multiplicative group of complex numbers, and also maximal
-subgroups
of the additive group of rational numbers modulo 1.
The ring of endomorphisms of a group of type
is isomorphic to the ring of
-adic
integers. A quasi-cyclic group coincides with its
Frattini subgroup.