Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
has order but is not cyclic
Statement refuted
The order of a unit group need not be the order of one of its elements: has order but is not cyclic.
Facts & Assumptions
Given: The modulus .
The unit-group structure theorem gives the product of the prime-power factors (The unit group modulo is the product of its odd-prime cyclic factors and its explicit -power factor).
An element of finite order has its th power equal to the identity exactly when (If then iff is an integer multiple of , the powers are distinct, and has exactly elements; if has infinite order then only for ).
Counterexample
By [L1], , which has elements.
For every pair , [L2] gives and , so . Another application of [L2] shows its order divides . Thus no element has order and the group is not cyclic.
Depends on
- The unit group modulo $n$ is the product of its odd-prime cyclic factors and its explicit $2$-power factor
- If $\operatorname{ord}(g) = n$ then $g^{k} = e$ iff $k$ is an integer multiple of $n$, the powers $g^{0}, \dots, g^{n-1}$ are distinct, and $\langle g \rangle$ has exactly $n$ elements; if $g$ has infinite order then $g^{j} = g^{k}$ only for $j = k$
Used by
Nothing in the library uses this result yet.
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 96 results over 23 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
- William Stein, Elementary Number Theory, §2.5 (standard reference, not scraped)