The center $Z(G)$ of a group

Depends on

• Group and abelian group

Used by

• A noncentral element of an extraspecial p-group has centraliser of index p Corollary
• An extraspecial p-group has order p¹⁺²ⁿ for some n≥1 Corollary
• An extraspecial p-group is nilpotent of class exactly two and its derived subgroup has order p Corollary
• An extraspecial p-group is the product of two maximal abelian subgroups meeting in its centre Corollary
• An extraspecial p-group of order p¹⁺²ⁿ has generator rank 2n 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
• G/Z(G)congInn(G) Corollary
• The center of a free product with at least two nontrivial factors is trivial Corollary
• The centre of an extraspecial p-group has no complement Corollary
• Z(Sₙ) is trivial for n≥3 Corollary
• For odd p, a direct product of two Heisenberg groups is special with centre of order p², hence not extraspecial Counterexample
• S₃ has trivial center, so the finite p-group hypothesis in the nontrivial-center theorem is necessary Counterexample
• Complete group Definition
• Internal central products of a finite family of subgroups Definition
• Special and extraspecial p-groups Definition
• The central product G∘_α H of two groups along an isomorphism of central subgroups Definition
• The centralizer C_G(H) of a subgroup Definition
• The commutator pairing of an extraspecial p-group relative to a chosen generator of its centre Definition
• The square map of an extraspecial 2-group relative to a chosen generator of its centre Definition
• The upper central series Definition
• Every inner automorphism of an abelian group is the identity Example
• For an abelian group G, Z(G)=G and [G,G]={e} Example
• FALSE for odd p: the scalar-valued commutator pairing needs no choice of a central generator False statement
• FALSE: every special p-group is extraspecial False statement
• 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
• Central factors are equivalent to adjacent commutator containments Lemma
• Commutator identities in a group whose derived subgroup is central Lemma
• 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
• If G/Z(G) is cyclic, then G is abelian Lemma
• In a group with central derived subgroup, (xy)ⁿ=[y,x]^C(n, 2)xⁿyⁿ Lemma
• Q₈∘ Q₈ congDih(C₄)circDih(C₄) Lemma
• The center of a group is a normal subgroup Lemma
• The commutator pairing is well defined on the central quotient, is bilinear over Fₚ, and is alternating Lemma
• The commutator pairing of an extraspecial p-group has trivial radical Lemma
• The identified subgroup used to form a central product is central, hence normal 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
• An automorphism of an extraspecial p-group acting trivially on its Frattini quotient is inner Proposition
• Dih(C₄) and Q₈ are extraspecial of order 8, with six and two solutions of x²=1 respectively Proposition

…and 22 more results.

Dependency tree · two levels

6 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

• Encyclopedia of Mathematics, Characteristic subgroup (standard reference, not scraped)