ExampleConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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There is a unique nonabelian group of order $21$ , namely $C_{7} ⋊ C_{3}$ with multiplication by $2$

Example

Up to isomorphism, the unique nonabelian group of order $21$ is

$C_{7} ⋊ C_{3},$

where a generator of $C_{3}$ acts on $C_{7}$ by raising elements to the second power.

Facts & Assumptions

Given: The primes $3 < 7$ .

[L1]

When $p ∣ (q - 1)$ , there is exactly one nonabelian group of order $p q$ , namely the nontrivial product $C_{q} ⋊ C_{p}$ (Classification of groups of order $p q$ for primes $p < q$ ).

[L2]

$Aut (C_{7}) ≅ (Z / 7)^{\times}$ , with $[a]$ acting by the exponent $a$ ( $Aut (C_{n}) ≅ (Z / n Z)^{\times}$ ).

Verification

technique · direct

1.1

Since $3 ∣ 6$ , [L1] gives a unique nonabelian isomorphism type of order $21$ .

L1algebra

1.2

Modulo $7$ , one has $2^{3} \equiv 1$ while $2 \equiv̸ 1$ , so $[2]$ has order three in $(Z / 7)^{\times}$ . By [L2], it defines a nontrivial action of $C_{3}$ on $C_{7}$ .

L2algebra

2.1

The resulting semidirect product is therefore the unique nonabelian group from [L1].

step 1.1step 1.2L1∎

Depends on

Used by

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Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 93 results over 17 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

J. S. Milne, Group Theory (standard reference, not scraped)

There is a unique nonabelian group of order 2121, namely C7⋊C3C7​⋊C3​ with multiplication by 22