Subject Cosets and Lagrange Theorem Given G group have numbe

Subject: Cosets and Lagrange Theorem Given G group have number of finite subgroups. Prove that G finite.

Solution

Let G be a group with only finitely many subgroups.

We consider two cases:

i. G has an element g of infinite order: In this case, the cyclic subgroup generated by g is isomorphic to Z. We know that Z has infinitely many subgroups (the subgroups n Z are distinct for all natural numbers n). Thus, G has infinitely many subgroups, contradicting the assumption.

· ii. Every element in G has finite order: In this case, since G is a union of cyclic subgroups, each of which is finite, and since G has only finitely many subgroups, therefore, G is a finite union of finite subgroups, and thus, G is finite.