Subgroups of G1 {0, 1, 2, 3, 4, 5} Under Addition Modulo 6

When examining the group G1 {0, 1, 2, 3, 4, 5} under addition modulo 6, it is important to understand the concept of a subgroup. A subgroup is a subset of a group that is itself a group under the operation defined on the original group. In this context, we will delve into finding all subgroups of G1.

Understanding the Group G1

G1 {0, 1, 2, 3, 4, 5} is a cyclic group under addition modulo 6. This means that every element in G1 can be generated by repeatedly adding a single generator to itself, modulo 6. Specific to G1, the generator can be 1, as every element can be represented as 1 added to itself a certain number of times, modulo 6.

Divisors of the Group Order

The order of G1 is 6. The subgroups of a cyclic group are determined by the divisors of the group's order. The divisors of 6 are 1, 2, 3, and 6. Each of these divisors corresponds to a unique subgroup in G1.

Subgroups of G1

Trivial Subgroup (Order 1)

The trivial subgroup, denoted as {0}, is a subgroup of G1 that contains only the identity element under the operation of addition modulo 6. This subgroup is always present in any cyclic group.

Subgroup of Order 2

This subgroup is generated by the element 3. Under addition modulo 6, the subgroup can be written as {0, 3}. Here, 0 3 ≡ 3 (mod 6) and 3 3 ≡ 0 (mod 6), completing the subgroup.

Subgroup of Order 3

The subgroup of order 3 is generated by the element 2. The elements of this subgroup under addition modulo 6 are {0, 2, 4}. This can be verified as follows: 0 2 ≡ 2 (mod 6), 2 2 ≡ 4 (mod 6), and 4 2 ≡ 0 (mod 6).

Whole Group (Order 6)

The group G1 {0, 1, 2, 3, 4, 5} is the largest subgroup, containing all elements of G1 itself. This is the full cyclic group itself, generated by 1 (or any other generator).

Conclusion

In summary, the subgroups of G1 {0, 1, 2, 3, 4, 5} under addition modulo 6 are {0}, {0, 3}, {0, 2, 4}, and G1 itself. These subgroups are derived from the divisors of the group's order and serve as key examples of how subgroups are identified within a cyclic group.