Math 2.0

Group theory and generalizations Proposition

If G modulo its centre is cyclic then G is abelian

Builds on Centre of a groupQuotient groupCyclic group

Statement

Let GG be a group with centre Z(G)Z(G). If the quotient group G/Z(G)G/Z(G) is cyclic, then GG is abelian.

Remarks

Let gZ(G)gZ(G) generate G/Z(G)G/Z(G); then each element of GG is gkzg^k z with k∈ℤk \in \mathbb{Z} and z∈Z(G)z \in Z(G), and two such elements commute. In that case G=Z(G)G = Z(G), so in fact G/Z(G)G/Z(G) is trivial. A consequence is that G/Z(G)G/Z(G) is never cyclic of order greater than 11. For example, the centre of a group never has prime index.

Reading path

To understand this page from the ground up, read these 14 pages in order. The foundations come first.

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Group, concept, no evidence levelGroupconceptCentre of a group, definition, no evidence levelCentre of a groupdefinitionConjugate elements, definition, no evidence levelConjugate elementsdefinitionLaws of exponents in a group, statement, no evidence levelLaws of exponents in …statementSubgroup, definition, no evidence levelSubgroupdefinitionConjugate subgroup, definition, no evidence levelConjugate subgroupdefinitionCyclic subgroup generated by an element, definition, no evidence levelCyclic subgroup gener…definitionLeft coset, definition, no evidence levelLeft cosetdefinitionRight coset, definition, no evidence levelRight cosetdefinitionCyclic group, definition, no evidence levelCyclic groupdefinitionNormal subgroup, definition, no evidence levelNormal subgroupdefinitionNormality via left and right cosets, statement, no evidence levelNormality via left an…statementQuotient group, definition, no evidence levelQuotient groupdefinitionIf G modulo its centre is cyclic then G is abelian, statement, no evidence levelIf G modulo its centr…statement
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. Group concept
  2. Centre of a group definition
  3. Conjugate elements definition
  4. Laws of exponents in a group statement
  5. Subgroup definition
  6. Conjugate subgroup definition
  7. Cyclic subgroup generated by an element definition
  8. Left coset definition
  9. Right coset definition
  10. Cyclic group definition
  11. Normal subgroup definition
  12. Normality via left and right cosets statement
  13. Quotient group definition
  14. If G modulo its centre is cyclic then G is abelian statement

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Center_(group_theory), Examples
About this page
Tag0AH9Q6YNMT
URIm2://0AH9Q6YNMT
Kindstatement, proposition
WritingW0 stub
MSC20A05
StatusThis copy of the site has no build data. Run m2 status 0AH9Q6YNMT.
UsesCentre of a group, Quotient group, Cyclic group

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