Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
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Internal direct products of finitely many normal subgroups
Definition
Let be a group and let be normal subgroups, where . They form an internal direct product when they generate and, for each , The empty family is an internal direct product of the trivial group. For two subgroups of an abelian group this says and ; in additive notation one writes . Normal subgroups and generated subgroups are those of Normal subgroup: invariance under conjugation and The subgroup generated by a subset, the cyclic subgroup , and cyclic groups, and the comparison product is The external direct product with componentwise multiplication.
Depends on
Used by
- Indecomposable and decomposable nontrivial finite abelian groups Definition
- Internal central products of a finite family of subgroups Definition
- The Fitting subgroup F(G)=∏ₚ Oₚ(G) of a finite group Definition
- Complements of a maximal cyclic subgroup in Cₚ times Cₚ need not be unique Example
- The unit group modulo one hundred is isomorphic to C₂0 times C₂ Example
- The socle is characteristic and decomposes as a direct product of minimal normal subgroups Proposition
- A maximal-order cyclic subgroup splits off a finite abelian p-group Theorem
- Finite characteristically simple groups are direct products of isomorphic simple groups Theorem
- Internal direct products are external direct products, equivalently every element has a unique factorisation Theorem
Dependency tree · two levels
7 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)