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The commutator pairing is well defined on the central quotient, is bilinear over , and is alternating
Statement
Let be an extraspecial -group with and . The commutator pairing is well defined on : the value of does not depend on the representatives and . It is -bilinear,
and alternating: for every . Consequently .
Facts & Assumptions
Given: An extraspecial -group with of order , the quotient with its canonical -structure, and the pairing defined by .
The commutator pairing of relative to is the map determined by , where carries the canonical scalar action (The commutator pairing of an extraspecial -group relative to a chosen generator of its centre).
For the commutator is (Commutators and the commutator subgroup ).
The rule gives every elementary abelian -group its canonical -vector-space structure, with the group operation as vector addition and the identity as zero (An elementary abelian -group has a canonical -vector-space structure).
An extraspecial -group is nilpotent of class exactly two, so its derived subgroup is central, and every nonidentity commutator has order (An extraspecial -group is nilpotent of class exactly two and its derived subgroup has order ).
If then , , and for every integer (Commutator identities in a group whose derived subgroup is central).
For every prime , the operations of addition and multiplication on make it a field (For every prime , the two operations on make it a field).
Proof
The derived subgroup of is central, so the two expansion identities and the power identity are available for all elements of .
Alternation: , so .
If then with , and because is central makes ; the second variable is the same computation with the other expansion identity. So depends only on the two cosets.
Additivity in the first variable: , and exponents of are determined modulo ; the second variable is symmetric.
Compatibility with scalars: for an integer , , and both sides depend only on modulo because has order ; the scalar action on is , so this is exactly , and likewise in the second variable.
Expanding by steps 2.2 and 1.2 gives , so .
Remarks
Alternation is the primitive property and skew symmetry is derived from it, not the other way round. At the two are not interchangeable: there , so skew symmetry says only that the pairing is symmetric, and it is the vanishing of that carries content.
Depends on
- Commutator identities in a group whose derived subgroup is central
- An extraspecial $p$-group is nilpotent of class exactly two and its derived subgroup has order $p$
- The commutator pairing of an extraspecial $p$-group relative to a chosen generator of its centre
- An elementary abelian $p$-group has a canonical $\mathbb F_p$-vector-space structure
- For every prime $p$, the two operations on $\mathbb{Z}/p$ make it a field
- Commutators $[g,h]=ghg^{-1}h^{-1}$ and the commutator subgroup $[G,G]$
- Powers $g^{n}$: natural exponents in a monoid and integer exponents in a group, with $g^{0} = e$
- The center $Z(G)$ of a group
- The quotient group $G/N$ and coset product $(gN)(hN)=ghN$
Used by
- An extraspecial p-group is the product of two maximal abelian subgroups meeting in its centre Corollary
- The commutator pairings of Dih(C₄) and Q₈ are the same, while the groups are not isomorphic Example
- FALSE for odd p: the scalar-valued commutator pairing needs no choice of a central generator False statement
- A subgroup of the central quotient and its orthogonal complement have orders multiplying to the order of the quotient Lemma
- The commutator pairing of an extraspecial p-group has trivial radical 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
- In an extraspecial p-group of order p¹⁺²ⁿ every maximal abelian subgroup has order p¹⁺ⁿ Proposition
- The maximal elementary abelian subgroups of the two extraspecial groups of order 2¹⁺²ⁿ have orders 2ⁿ⁺¹ and 2ⁿ Proposition
- Every extraspecial p-group is an internal central product of nonabelian subgroups of order p³ Theorem
Cited to discharge well-definedness by The commutator pairing of an extraspecial p-group relative to a chosen generator of its centre.
Dependency tree · two levels
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Sources
- D. A. Craven, The Theory of p-Groups, Lemma 3.7 and §3.2 (standard reference, not scraped)
- M. van Beek, Topics in Finite p-Groups, §2.4 (standard reference, not scraped)