Math 2.0

Group theory and generalizations Proof

Proof that abelian groups are solvable

Builds on Abelian groupSolvable groupSubnormal seriesNormal subgroupQuotient group

Proof

  1. Let GG be an abelian group, and let G0={e}G_0 = \{e\} and G1=GG_1 = G.
  2. The subgroup {e}\{e\} is normal in GG, because geg−1=egeg^{-1} = e for each g∈Gg \in G.
  3. So {e}=G0⊴G1=G\{e\} = G_0 \trianglelefteq G_1 = G is a subnormal series of GG of length 11.
  4. Its only factor is the quotient group G/{e}G/\{e\}. For a,b∈Ga, b \in G we have (a{e})(b{e})=(ab){e}=(ba){e}=(b{e})(a{e})(a\{e\})(b\{e\}) = (ab)\{e\} = (ba)\{e\} = (b\{e\})(a\{e\}), because GG is abelian.
  5. So the factor G/{e}G/\{e\} is abelian, and GG is solvable by definition.

Idea

The shortest possible series, from the trivial group straight to GG, already works. Its single factor is a copy of GG, which is abelian.

Reading path

To understand this page from the ground up, read these 16 pages in order. The foundations come first. The path takes the proof Proof that a subgroup is normal iff its left and right cosets agree.

Show the path as a graph
Binary operation, definition, no evidence levelBinary operationdefinitionGroup, concept, no evidence levelGroupconceptCommutativity, definition, no evidence levelCommutativitydefinitionConjugate elements, definition, no evidence levelConjugate elementsdefinitionSubgroup, definition, no evidence levelSubgroupdefinitionAbelian group, definition, no evidence levelAbelian groupdefinitionConjugate subgroup, definition, no evidence levelConjugate subgroupdefinitionLeft coset, definition, no evidence levelLeft cosetdefinitionRight coset, definition, no evidence levelRight cosetdefinitionNormal subgroup, definition, no evidence levelNormal subgroupdefinitionProof that a subgroup is normal iff its left and right cosets agree, proof, no evidence levelProof that a subgroup…proofNormality via left and right cosets, statement, no evidence levelNormality via left an…statementQuotient group, definition, no evidence levelQuotient groupdefinitionSubnormal series, definition, no evidence levelSubnormal seriesdefinitionSolvable group, definition, no evidence levelSolvable groupdefinitionProof that abelian groups are solvable, proof, no evidence levelProof that abelian gr…proof
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Binary operation definition
  2. Group concept
  3. Commutativity definition
  4. Conjugate elements definition
  5. Subgroup definition
  6. Abelian group definition
  7. Conjugate subgroup definition
  8. Left coset definition
  9. Right coset definition
  10. Normal subgroup definition
  11. Proof that a subgroup is normal iff its left and right cosets agree proof
  12. Normality via left and right cosets statement
  13. Quotient group definition
  14. Subnormal series definition
  15. Solvable group definition
  16. Proof that abelian groups are solvable proof

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Solvable_group, Abelian groups
About this page
TagQAWKASHQBY
URIm2://QAWKASHQBY
Kindproof
WritingW0 stub
MSC20F16
ProvesAbelian groups are solvable
UsesAbelian group, Solvable group, Subnormal series, Normal subgroup, Quotient group
Changed4 November 2026

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