Classification of Subgroups of Cyclic Groups A cyclic group is a group that is generated by a single element. Cyclic groups are classified based on their or...
Classification of Subgroups of Cyclic Groups
A cyclic group is a group that is generated by a single element. Cyclic groups are classified based on their order, which is the number of elements in the group.
Order of a Cyclic Group:
The order of a cyclic group is the number of elements in the group. The order of a cyclic group is always a prime number.
Classification Based on Order:
Cyclic groups are classified into the following groups based on their order:
Cyclic groups of order 1: A cyclic group of order 1 is a trivial group containing only the identity element.
Cyclic groups of order 2: Cyclic groups of order 2 are isomorphic to the cyclic group of order 4. They have two subgroups, the identity subgroup and the cyclic subgroup generated by the generator 2.
Cyclic groups of order 3: Cyclic groups of order 3 are isomorphic to the cyclic group of order 6. They have three subgroups: the identity subgroup, the cyclic subgroup generated by the generator 3, and the cyclic subgroup generated by the generator 6.
Cyclic groups of order 4: Cyclic groups of order 4 are isomorphic to the cyclic group of order 8. They have four subgroups: the identity subgroup, the cyclic subgroup generated by the generator 2, the cyclic subgroup generated by the generator 3, and the cyclic subgroup generated by the generator 4.
Cyclic groups of order 6: Cyclic groups of order 6 are isomorphic to the cyclic group of order 12. They have six subgroups: the identity subgroup, the cyclic subgroup generated by the generator 2, the cyclic subgroup generated by the generator 3, the cyclic subgroup generated by the generator 4, and the two cyclic subgroups generated by the generators 5 and 6.
Examples:
The cyclic group of order 1 is the trivial group {e}.
The cyclic group of order 2 is isomorphic to the cyclic group of order 4, which has two subgroups: the identity subgroup and the cyclic subgroup generated by the generator 2.
The cyclic group of order 3 is isomorphic to the cyclic group of order 6, which has three subgroups: the identity subgroup, the cyclic subgroup generated by the generator 3, and the cyclic subgroup generated by the generator 6