Write do n all the subgroups of (𝑍6, +6) and (𝑍8, +8

Sure, let's find all the subgroups of (𝑍6, +6) and (𝑍8, +8).

1. Subgroups of (𝑍6, +6):

The subgroups of (𝑍6, +6) are generated by the divisors of 6. The divisors of 6 are 1, 2, 3, and 6. So, the subgroups are:

• <0> = {0}
• <1> = {0, 1, 2, 3, 4, 5} = 𝑍6
• <2> = {0, 2, 4}
• <3> = {0, 3}
• <6> = {0}

2. Subgroups of (𝑍8, +8):

The subgroups of (𝑍8, +8) are generated by the divisors of 8. The divisors of 8 are 1, 2, 4, and 8. So, the subgroups are:

• <0> = {0}
• <1> = {0, 1, 2, 3, 4, 5, 6, 7} = 𝑍8
• <2> = {0, 2, 4, 6}
• <4> = {0, 4}
• <8> = {0}

Note: The notation <n> denotes the subgroup generated by n.