formulate the theorem about subgroups of finite cyclic group

formulate the theorem about subgroups of finite cyclic group

Solution

The theorems about subgroups of a cyclic group:

We state here, three theorems which are applicable on finite cyclic groups:

Theorem 1. The subgroup of a cyclic group is itself a cyclic group.

For example: The subgroups of the finite cyclic group

{a, a², a³, a⁴, a⁵, a⁶=e} are {a³, a⁶=e}, and {a², a⁴, a⁶=e}

which are cyclic groups of order 2 and 3.

Theorem 2. If H is a subgroup of a cyclic group G and if G is of order n > 0 then H is of order n/m and m divides n.

Theorem 3. If G is a finite cyclic group of order n and m is a divisor of n, then there exists one and only one subgroup H of order m, which is also cyclic.