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.
The th-power subgroup
Definition
For a group and a prime (Prime and composite integers: is prime when and its only positive divisors are and ), the th-power subgroup is
where powers are those of Powers : natural exponents in a monoid and integer exponents in a group, with and the generated subgroup is that of The subgroup generated by a subset, the cyclic subgroup , and cyclic groups. This notation denotes the subgroup generated by the powers, not merely the set of powers, which need not itself be a subgroup in an arbitrary group.
Depends on
- The subgroup $\langle S \rangle$ generated by a subset, the cyclic subgroup $\langle g \rangle$, and cyclic groups
- Powers $g^{n}$: natural exponents in a monoid and integer exponents in a group, with $g^{0} = e$
- Prime and composite integers: $p$ is prime when $p > 1$ and its only positive divisors are $1$ and $p$
Used by
- Φ(P)=P² for a finite 2-group Corollary
- Φ(P)=P'Pᵖ for a finite p-group Theorem
Dependency tree · two levels
24 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- D. A. Craven, The Theory of p-Groups, Definition 2.26 (standard reference, not scraped)
- K. Conrad, Generating Sets, §6 (standard reference, not scraped)