How to Use
Enter n:
- Table: ord(a) for all a ∈ (Z/nZ)*
- Groups: Elements by order
- Structure: Cyclic or not?
Group Structure
- Prime p: cyclic Z/(p−1)
- 2: trivial, 4: Z/2
- 2ᵏ (k≥3): Z/2 × Z/2ᵏ⁻²
- pᵏ (odd): cyclic Z/φ(pᵏ)
- General n: product via CRT
Order Distribution
For each d|φ(n), exactly φ(d) elements have order d (in cyclic groups). Non-cyclic groups have different distributions. The number of primitive roots is φ(φ(n)) (when they exist).
Step-by-Step Instructions
- 1Enter n.
- 2View order table.
- 3See grouping.
- 4Identify structure.
- 5Count by order.
Group Element Order Table — Frequently Asked Questions
When is (Z/nZ)* cyclic?+
Iff n = 1, 2, 4, pᵏ, or 2pᵏ (p odd prime). This is exactly when primitive roots exist. For n=8: (Z/8Z)* = {1,3,5,7} ≅ Z/2 × Z/2 (not cyclic, no element has order 4=φ(8)).
How does CRT decompose the group?+
If n=n₁n₂ with gcd(n₁,n₂)=1, then (Z/nZ)* ≅ (Z/n₁Z)* × (Z/n₂Z)*. Orders in the product: ord(a,b) = lcm(ord(a), ord(b)). This explains why composite moduli often lack primitive roots.
What determines the largest order?+
The largest order in (Z/nZ)* is the Carmichael function λ(n). For cyclic groups: λ(n)=φ(n). For non-cyclic: λ(n)<φ(n). Example: λ(8)=2 < φ(8)=4. λ(n) = lcm of λ over prime power factors.
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