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.
Elementary abelian -groups
Definition
An elementary abelian -group is a finite abelian -group in which every nonidentity element has order ; the trivial group is permitted (A finite -group has order for a prime and some , Group and abelian group, The order of a finite group and the order of an element, with when no positive power of is the identity).
Depends on
Used by
- A finite p-group has trivial Frattini subgroup exactly when it is elementary abelian Corollary
- The centre of an extraspecial p-group has no complement Corollary
- For odd p, a direct product of two Heisenberg groups is special with centre of order p², hence not extraspecial Counterexample
- Special and extraspecial p-groups Definition
- The commutator pairing of an extraspecial p-group relative to a chosen generator of its centre Definition
- A product formula for the number of square roots of the identity in a central product of extraspecial 2-groups Lemma
- A subgroup of the central quotient and its orthogonal complement have orders multiplying to the order of the quotient Lemma
- An elementary abelian p-group has a canonical Fₚ-vector-space structure Lemma
- The square map is well defined on the central quotient and satisfies q(x̄ȳ)=q(x̄)+q(ȳ)+b(x̄,ȳ) Lemma
- Two elements of an extraspecial p-group with nontrivial commutator generate an extraspecial subgroup of order p³ Lemma
- A unique abelian minimal normal subgroup gives affine type Proposition
- Dih(C₄) and Q₈ are extraspecial of order 8, with six and two solutions of x²=1 respectively Proposition
- The Heisenberg group of order p³ is extraspecial, and for odd p it has exponent p Proposition
- The maximal elementary abelian subgroups of the two extraspecial groups of order 2¹⁺²ⁿ have orders 2ⁿ⁺¹ and 2ⁿ Proposition
- The modular group of order p³ is extraspecial, of exponent p² when p is odd Proposition
- Three equivalent descriptions of an extraspecial p-group Proposition
- A central product of extraspecial p-groups identified along their centres is extraspecial Theorem
- A nonabelian group of order p³ is extraspecial Theorem
- Every extraspecial p-group is an internal central product of nonabelian subgroups of order p³ Theorem
- The Frattini quotient is the largest elementary abelian quotient of a finite p-group Theorem
Dependency tree · two levels
19 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 2.6 (standard reference, not scraped)