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 quotient group of an abelian group is abelian
Statement
If is abelian and , then is abelian.
Facts & Assumptions
Given: An abelian group and a normal subgroup .
In the quotient group, (For , the cosets form a group with identity and inverse ).
A group is abelian when for all of its elements (Group and abelian group).
Proof
For arbitrary cosets , commutativity in gives .
Hence every two elements of commute, so is abelian.
Depends on
Used by
- Converse of Lagrange for finite abelian groups: every divisor occurs as a subgroup order Corollary
- A nontrivial finite abelian p-group with a unique subgroup of order p is cyclic Lemma
- A p-primary component has the full p-power order and is the unique subgroup of that order Theorem
- Cauchy's theorem for finite abelian groups Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 18 results over 15 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.
Sources
- T. W. Judson, Abstract Algebra: Theory and Applications, Normal Subgroups and Factor Groups, Exercises (standard reference, not scraped)