The external direct product $G\times H$ with componentwise multiplication

Depends on

• Group and abelian group

Used by

• Composition factors do not determine the extension Counterexample
• For odd p, a direct product of two Heisenberg groups is special with centre of order p², hence not extraspecial Counterexample
• Elementary-divisor data for a finite abelian group Definition
• Internal direct products of finitely many normal subgroups Definition
• Invariant-factor data for a finite abelian group Definition
• The central product G∘_α H of two groups along an isomorphism of central subgroups Definition
• In Zp squared, topological generation is detected by the Frattini quotient coordinates Example
• The canonical surjection from a free product to the direct product of its factors Example
• The central product of two cyclic groups of order four along their subgroups of order two is abelian of order eight Example
• The finite Heisenberg group is the unique Sylow p-subgroup of its coordinate upper-triangular group Example
• The split extension C₂ × C₂ of C₂ by C₂ is direct Example
• The torsion-free reflection of the integers direct sum a finite cyclic group Example
• FALSE: every finite group is a direct product of cyclic prime-power groups False statement
• A direct product of two finite cyclic groups is cyclic if and only if their orders are coprime Lemma
• A product formula for the number of square roots of the identity in a central product of extraspecial 2-groups Lemma
• The identified subgroup used to form a central product is central, hence normal Lemma
• Order, centre and derived subgroup of a central product Proposition
• The canonical semidirect decomposition is an internal direct product if and only if the defining action is trivial Proposition
• The profinite completion of the integers is the direct product of the p-adic integer groups over all primes Proposition
• The two canonical maps into a central product are injective homomorphisms whose images commute, generate it, and meet in the identified centre Proposition
• Φ(P× Q)=Φ(P)×Φ(Q) for finite p-groups Proposition
• Every finite abelian group is the Galois group of some finite Galois extension of ℚ Theorem
• For prime p, |Aut((ℤ/p)×(ℤ/p))|=(p²-1)(p²-p) Theorem
• G× H is a group with identity (e_G,e_H), coordinatewise inverses, and homomorphic coordinate projections Theorem
• Homomorphisms out of a central product Theorem
• Internal central products are the images of external ones Theorem
• π₁(X× Y,(x₀,y₀))≅π₁(X,x₀)×π₁(Y,y₀) Theorem

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

• Sharifi, Abstract Algebra, direct products (standard reference, not scraped)