Finitely many subgroups iff finite
From Groupprops
Contents
Statement
The following are equivalent for a group:
- The group is a Finite group (?), i.e., its order (the number of elements in its underlying set) is finite.
- The group has only finitely many subgroups.
Facts used
Proof
Finite implies finitely many subgroups
Given: A finite group .
To prove: has only finitely many subgroups.
Proof: Since is finite, the set of subsets of
is finite. Since subgroups are subsets satisfying additional conditions, the set of subgroups of
is also finite.
Finitely many subgroups implies finite
Given: A group with only finitely many subgroups.
To prove: is finite.
Proof: We consider two cases:
-
has an element
of infinite order: In this case, the cyclic subgroup generated by
is isomorphic to
.
has infinitely many subgroups (the subgroups
are distinct for all natural numbers
). Thus,
has infinitely many subgroups, contradicting the assumption.
- Every element in
has finite order: In this case, by fact (1),
is a union of cyclic subgroups, each of which is finite. Since
has only finitely many subgroups,
is a finite union of finite subgroups, and thus,
is finite.