The Circle That Returns

At a glance

• Part 1: The six roots are 𝜁0,𝜁1,𝜁2,𝜁3,𝜁4,𝜁5, at angles 0∘,60∘,120∘,180∘,240∘,300∘.
• Part 2: The generators are 𝜁 and 𝜁5. Their cycles have length 6.
• Part 3: The six roots form a group under multiplication with identity 𝜁0 =1, and the group is cyclic because the powers of 𝜁 produce every root.

Key idea: turns become exponents

Multiplication adds angles, so powers of one sixth-turn walk around a regular hexagon. Reducing exponents modulo 6 records that returning through a full turn changes nothing.

1. Find all sixth roots

Counterclockwise from angle 0∘, the vertices are

Every listed point works because

To prove there are no other roots, first note that 𝑧 =0 cannot satisfy 𝑧6 =1. Write any possible solution as 𝑧 =𝑟𝑅(𝜃) with 𝑟 >0. Then

The magnitudes must agree, so 𝑟6 =1. Since 𝑟 ≥0, this forces 𝑟 =1.

The angles must agree up to full turns, so

Therefore

These are exactly the six labeled vertices. Thus the list contains all and only the solutions of 𝑧6 =1.

2. Trace the cycles

Repeated multiplication by 𝜁𝑗 adds 𝑗 to the exponent each time, with exponents reduced modulo 6.

Hence the generators are

Another facet: predict the cycle length

Repeated multiplication by 𝜁𝑗 visits exponents

The first return occurs at the smallest positive whole number 𝑛 for which 6 divides 𝑛𝑗. Equivalently, the cycle length is

For 𝑗 =1,2,3,4,5, this gives 6,3,2,3,6, exactly the return times in the table. A root visits all six positions precisely when its exponent shares no factor greater than 1 with 6. That is why 𝜁 and 𝜁5 are the generators.

3. Prove the group structure

Let

Closure: If 𝜁𝑎,𝜁𝑏 ∈𝑆, then

Reducing 𝑎 +𝑏 modulo 6 gives one of the exponents 0,1,2,3,4,5, so the product remains in 𝑆.

Associativity: This is inherited from complex multiplication.

Identity: The identity is

Inverses: The inverse of 𝜁0 is itself. For 1 ≤𝑘 ≤5,

The inverse pairs are

All four group properties hold. The group is cyclic because the powers

of the single element 𝜁 exhaust the group.

Check

The cycle table agrees with the group proof: the identity occurs at exponent 0, each row eventually returns to that identity, and the two generator rows visit every group member before returning.