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.
is not cyclic because every element squares to
Example
is not cyclic: every element has square .
Facts & Assumptions
Given: The unit group .
The standard representatives of units are exactly the integers with and (For , every class in has one representative with , so ; while is in bijection with , For , is a unit if and only if , The unit group and Euler's totient for ).
A finite group is cyclic when one element generates it, and an element generates a finite group exactly when its order is the group cardinality (The subgroup generated by a subset, the cyclic subgroup , and cyclic groups, The order of a finite group and the order of an element, with when no positive power of is the identity, If then iff is an integer multiple of , the powers are distinct, and has exactly elements; if has infinite order then only for ).
Natural powers in a group satisfy and (Powers : natural exponents in a monoid and integer exponents in a group, with ).
Equality of residue classes is congruence of representatives (The congruence class and the quotient set ), and congruence means divisibility of their difference (Congruence modulo an integer: when , including the moduli and ).
Verification
The odd standard representatives are precisely those coprime to , giving the displayed unit group by [L1].
Their squares are , , , and . Hence every element has order at most .
The group has cardinality by step 1.1, but no element has order by step 1.2. Therefore no element generates it, and [L2] shows that it is not cyclic.
Depends on
- For $n\ge1$, $[a]_n$ is a unit if and only if $\gcd(a,n)=1$
- The unit group $(\mathbb{Z}/n)^\times$ and Euler's totient $\varphi(n)=\lvert(\mathbb{Z}/n)^\times\rvert$ for $n\ge1$
- Powers $g^{n}$: natural exponents in a monoid and integer exponents in a group, with $g^{0} = e$
- The order $|G|$ of a finite group and the order $\operatorname{ord}(g)$ of an element, with $\operatorname{ord}(g) = \infty$ when no positive power of $g$ is the identity
- The subgroup $\langle S \rangle$ generated by a subset, the cyclic subgroup $\langle g \rangle$, and cyclic groups
- 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 $j = k$
- For $n\ge 1$, every class in $\mathbb{Z}/n$ has one representative $r$ with $0\le r<n$, so $\lvert\mathbb{Z}/n\rvert=n$; while $\mathbb{Z}/0$ is in bijection with $\mathbb{Z}$
- The congruence class $[a]_n$ and the quotient set $\mathbb{Z}/n$
- Congruence modulo an integer: $a\equiv b\pmod n$ when $n\mid(a-b)$, including the moduli $0$ and $1$
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: 100 results over 27 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
- K. Conrad, Modular Arithmetic (standard reference, not scraped)
- University of Washington, Solutions for Assignment 2 (standard reference, not scraped)