Math 2.0

Group theory and generalizations Definition

Upper central series

Builds on Centre of a groupQuotient groupCorrespondence theorem

Definition

Let GG be a group. The upper central series of GG is the sequence of normal subgroups Z0={e}Z_0 = \{e\} and Zi+1={x∈G:[x,y]∈Zi for all y∈G}Z_{i+1} = \{x \in G : [x, y] \in Z_i \text{ for all } y \in G\} for i≥0i \geq 0.

Remarks

The first term Z1Z_1 is the centre Z(G)Z(G), and Z2Z_2 is called the second centre. For each ii, Zi+1/ZiZ_{i+1}/Z_i is the centre of the quotient group G/ZiG/Z_i. For the quaternion group Q8Q_8, the series is {1}⊆{1,−1}⊆Q8\{1\} \subseteq \{1, -1\} \subseteq Q_8. A group with trivial centre, such as S3S_3, has Zi={e}Z_i = \{e\} for each ii.

Reading path

To understand this page from the ground up, read these 16 pages in order. The foundations come first.

Show the path as a graph
Group, concept, no evidence levelGroupconceptCentre of a group, definition, no evidence levelCentre of a groupdefinitionConjugate elements, definition, no evidence levelConjugate elementsdefinitionSubgroup, definition, no evidence levelSubgroupdefinitionConjugate subgroup, definition, no evidence levelConjugate subgroupdefinitionLeft coset, definition, no evidence levelLeft cosetdefinitionRight coset, definition, no evidence levelRight cosetdefinitionSubgroup criterion, statement, no evidence levelSubgroup criterionstatementNormal subgroup, definition, no evidence levelNormal subgroupdefinitionIntersection of subgroups is a subgroup, statement, no evidence levelIntersection of subgr…statementNormality via left and right cosets, statement, no evidence levelNormality via left an…statementSubgroup generated by a subset, definition, no evidence levelSubgroup generated by…definitionQuotient group, definition, no evidence levelQuotient groupdefinitionLattice of subgroups, concept, no evidence levelLattice of subgroupsconceptCorrespondence theorem, statement, no evidence levelCorrespondence theoremstatementUpper central series, definition, no evidence levelUpper central seriesdefinition
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. Subgroup definition
  5. Conjugate subgroup definition
  6. Left coset definition
  7. Right coset definition
  8. Subgroup criterion statement
  9. Normal subgroup definition
  10. Intersection of subgroups is a subgroup statement
  11. Normality via left and right cosets statement
  12. Subgroup generated by a subset definition
  13. Quotient group definition
  14. Lattice of subgroups concept
  15. Correspondence theorem statement
  16. Upper central series definition

Where this leads

2 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Central_series, Upper central series
About this page
Tag0DJ9Y9V610
URIm2://0DJ9Y9V610
Aliasesascending-central-series
Kinddefinition
WritingW0 stub
MSC20F14
DefinesUpper central series
UsesCentre of a group, Quotient group, Correspondence theorem

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

Used by