Question

Show that any infinite group has proper subgroups.​

Answer 1

All subgroups of Zp∞ are finite Let H be a proper subgroup of Zp∞. We prove that H is equal to one of the Zpn for n≥0. If the set of the orders of elements of H is infinite, then for all element z∈Zp∞ of order pk, there would exist an element z′∈H of order pk′>pk. Hence H would contain Zp′ and z∈H.