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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Every subgroup of an abelian group is normal
Statement
Every subgroup of an abelian group is normal.
Facts & Assumptions
Given: An abelian group and a subgroup .
A group is abelian when for all of its elements (Group and abelian group).
A subgroup is normal if and only if for every (Equivalent characterisations of a normal subgroup by conjugates and left and right cosets).
Proof
For every , commutativity gives .
Hence by the coset characterisation of normality.
Depends on
Used by
- Every connected covering of the circle is regular Corollary
- The quotient ring R/I with (r+I)(s+I)=rs+I Definition
- Abelian groups and ℤ-modules have the same objects and morphisms Proposition
- For every n∈ℕ, the congruence-class group (ℤ/n,+) is the quotient group (ℤ,+)/nℤ Proposition
- A ring homomorphism whose kernel contains a two-sided ideal factors uniquely through the quotient ring Theorem
Dependency tree · two levels
9 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, Normal subgroup (standard reference, not scraped)