ExampleConstruction: AI-generatedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-16

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The unit group modulo $240$ decomposes as $C_{2}^{2} \times C_{4}^{2}$

Example

The unit group modulo $240$ has the decomposition

$(Z / 240)^{\times} ≅ C_{2}^{2} \times C_{4}^{2} .$

Facts & Assumptions

Given: The factorisation $240 = 16 \cdot 3 \cdot 5$ .

[L1]

The unit-group structure theorem decomposes a modulus into its prime-power unit groups (The unit group modulo $n$ is the product of its odd-prime cyclic factors and its explicit $2$ -power factor).

[L2]

Euler's totient is multiplicative on coprime positive arguments (Euler's totient is multiplicative: $\gcd (m, n) = 1$ implies $φ (m n) = φ (m) φ (n)$ for positive $m, n$ ).

Verification

technique · direct

1.1

By [L1], $U (16) ≅ C_{2} \times C_{4}$ , $U (3) ≅ C_{2}$ , and $U (5) ≅ C_{4}$ .

L1algebra

2.1

Their product is $C_{2}^{2} \times C_{4}^{2}$ . Its order is $2^{2} \cdot 4^{2} = 64$ , agreeing with $φ (240) = φ (16) φ (3) φ (5) = 8 \cdot 2 \cdot 4 = 64$ by [L2].

step 1.1L2algebra∎

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The unit group modulo 240240 decomposes as C22×C42C22​×C42​