What are the Possible Subgroups of Z6 Under Addition?

The possible subgroups of Z6 under addition are {0}, {0, 2, 4}, {0, 3}, and {0, 1, 2, 3, 4, 5}.

Z6, also known as the cyclic group of order 6, consists of the set of integers {0, 1, 2, 3, 4, 5} under the operation of addition modulo 6. Understanding its subgroups involves recognizing which subsets of Z6 form a group under the same operation.

Understanding Subgroups of Z6

Every subgroup of a cyclic group is also cyclic. This means each subgroup of Z6 can be generated by a single element from Z6. The order of a subgroup must be a divisor of the order of the main group (Lagrange's Theorem). Since Z6 has an order of 6, its subgroups can only have orders that are divisors of 6, which are 1, 2, 3, and 6.

Let's examine the subgroups generated by each element in Z6:

• Generated by 0 (⟨0⟩):

  • Starting with 0 and repeatedly adding 0 (mod 6) gives only {0}.
  • Subgroup: {0}
  • Order: 1 (trivial subgroup)

• Generated by 1 (⟨1⟩):

  • 1 (mod 6) = 1
  • 1 + 1 = 2 (mod 6) = 2
  • 2 + 1 = 3 (mod 6) = 3
  • 3 + 1 = 4 (mod 6) = 4
  • 4 + 1 = 5 (mod 6) = 5
  • 5 + 1 = 6 (mod 6) = 0
  • Subgroup: {0, 1, 2, 3, 4, 5} = Z6
  • Order: 6 (the group itself, considered a non-proper subgroup)

• Generated by 2 (⟨2⟩):

  • 2 (mod 6) = 2
  • 2 + 2 = 4 (mod 6) = 4
  • 4 + 2 = 6 (mod 6) = 0
  • Subgroup: {0, 2, 4}
  • Order: 3

• Generated by 3 (⟨3⟩):

  • 3 (mod 6) = 3
  • 3 + 3 = 6 (mod 6) = 0
  • Subgroup: {0, 3}
  • Order: 2

• Generated by 4 (⟨4⟩):

  • 4 (mod 6) = 4
  • 4 + 4 = 8 (mod 6) = 2
  • 2 + 4 = 6 (mod 6) = 0
  • Subgroup: {0, 2, 4} (This is the same subgroup as ⟨2⟩)

• Generated by 5 (⟨5⟩):

  • 5 (mod 6) = 5
  • 5 + 5 = 10 (mod 6) = 4
  • 4 + 5 = 9 (mod 6) = 3
  • 3 + 5 = 8 (mod 6) = 2
  • 2 + 5 = 7 (mod 6) = 1
  • 1 + 5 = 6 (mod 6) = 0
  • Subgroup: {0, 1, 2, 3, 4, 5} = Z6 (This is the same subgroup as ⟨1⟩)

Summary of Subgroups

The unique subgroups of Z6, listed in increasing order of their size, are:

These four are the only possible subgroups of Z6 under addition. They correspond directly to the divisors of 6 (1, 2, 3, and 6), with each divisor representing the order of a unique subgroup.