Alphabeta MathHow statement and proof provenance work
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A finite -group has order for a prime and some
Definition
Let be a prime natural number (Prime and composite integers: is prime when and its only positive divisors are and ). A finite -group is a finite group (Group and abelian group, The cardinality of a finite set) whose order has the form
for some , with natural exponentiation as in Exponentiation of natural numbers, , and its agreement with the integer power in . The case permits the trivial group. A finite -group is nontrivial exactly when .
Depends on
Used by
- Every group of order p², for prime p, is abelian Corollary
- Every subgroup of index p in a finite p-group is normal Corollary
- Elementary abelian p-groups Definition
- p-elementary and p-hyperelementary finite groups Definition
- Special and extraspecial p-groups Definition
- Sylow p-subgroups of a finite group Definition
- The p-core Oₚ(G) as the largest normal p-subgroup Definition
- A finite product of normal p-subgroups is a normal p-subgroup Lemma
- Every subgroup of a finite p-group has order a power of p Lemma
- Every conjugacy class of an extraspecial p-group outside the centre has exactly p elements Proposition
- 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
- 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
- Cauchy's theorem: if a prime p divides |G|, then G has an element of order p Theorem
- Every nontrivial finite p-group has nontrivial center, in fact p divides |Z(P)| Theorem
- Every nontrivial normal subgroup of a finite p-group meets the center nontrivially Theorem
- If a finite p-group P acts on a finite set X, then |X|≡|X^P| (mod p) Theorem
Dependency tree · two levels
40 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
- K. Conrad, Group Actions, Section 4 (standard reference, not scraped)