Alphabeta MathHow statement and proof provenance work
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The exponent of a finite group
Definition
For a finite group , its exponent is The set is nonempty by for every element of a finite group , and The well-ordering principle gives its least member; powers use Powers : natural exponents in a monoid and integer exponents in a group, with . Thus the definition is well-defined. For the trivial group .
Depends on
Used by
- An extraspecial group of odd order has exponent p or p², and an extraspecial 2-group has exponent 4 Corollary
- Every finite subgroup of the unit group of an integral domain is cyclic Corollary
- For an odd prime p, the p-th power map is a homomorphism on a finite group whose derived subgroup is central of exponent dividing p Corollary
- If gcd(a,n)=1, then a^λ(n)≡1 (mod n) Corollary
- Invariant factors determine the order and exponent of a finite abelian group Corollary
- Carmichael's function λ(n) as the exponent of (ℤ/nℤ)^× Definition
- Plus and minus type of an extraspecial p-group Definition
- The Heisenberg group of order 27 has exponent 3 and thirteen subgroups of order 3 Example
- The modular group of order 27 has exponent 9 and exactly three cyclic subgroups of order 9 Example
- FALSE: for each n≥1 there is exactly one extraspecial group of order p¹⁺²ⁿ up to isomorphism False statement
- For odd p, a central product of two modular groups of order p³ is a central product of a modular group with a Heisenberg group Lemma
- The Heisenberg group of order p³ is extraspecial, and for odd p it has exponent p Proposition
- The modular group of order p³ is extraspecial, of exponent p² when p is odd Proposition
- Brauer's cyclotomic criterion for splitting fields is recorded here only as an external theorem Remark
- For each prime there are exactly two nonabelian groups of order p³ up to isomorphism Theorem
- For odd p and each n≥1 there are exactly two extraspecial groups of order p¹⁺²ⁿ, distinguished by their exponent Theorem
Dependency tree · two levels
25 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
- Keith Conrad, Decomposition of Finite Abelian Groups, §§1-4 (standard reference, not scraped)
- Richard Elman, Lectures on Abstract Algebra, Ch. 14 (standard reference, not scraped)