Cyclic group of order 5
A cyclic group with 5 one-dimensional ordinary complex irreducibles.
Reference properties
Representation scope
The complete list of ordinary complex irrep dimensions is given above. Labels ρ1, ρ2, … follow the cited GroupNames character-table row order; they are local labels, not dimension-based identities. [sources]
Real, Complex and Quaternionic refer to Frobenius–Schur type (+1, 0, −1). The linked character table provides additional data; this entry does not supply matrices or a modular representation catalogue. [sources]
| Irrep label | Complex dimension | Reality type |
|---|---|---|
| ρ1 | 1 | Real |
| ρ2 | 1 | Complex |
| ρ3 | 1 | Complex |
| ρ4 | 1 | Complex |
| ρ5 | 1 | Complex |
Structure and conventions
For a generator a, χₖ(a) = exp(2πik/5); source row labels need not be ordered by k. [sources]
Names and aliases
Z5 · Z/5Z · ℤ5
Related objects and maps
- Family
- Family
Claims and sources
Each assertion has a stable identifier within this record and a source locator or derivation. Cite the dataset edition when referring to a corrected or historical claim.
Definition
The cyclic group generated by a with a^5 = 1 and exact order 5.
- GroupNames: C₅Heading; Aliases; Generators and relationsOpen source
group:c-5#definitionOrdinary complex irreducibles
1, 1, 1, 1, 1
Assumptions: Characteristic zero · Finite-dimensional irreducibles over C · Dimensions listed with multiplicity
- GroupNames: C₅Character table: identity column and real/complex/symplectic annotationsOpen source
- GAP · Characters and symmetrizationsIrr and section 71.12-5 IndicatorOpen source
group:c-5#irrepsGAP SmallGroups ID
SmallGroup(5,1)
- GroupNames: C₅Aliases; GAP/Magma code by IDOpen source
group:c-5#smallgroups-idRepresentation scope
The complete list of ordinary complex irrep dimensions is given above. Labels ρ1, ρ2, … follow the cited GroupNames character-table row order; they are local labels, not dimension-based identities.
- GroupNames: C₅Character tableOpen source
group:c-5#representations-1Representation scope
Real, Complex and Quaternionic refer to Frobenius–Schur type (+1, 0, −1). The linked character table provides additional data; this entry does not supply matrices or a modular representation catalogue.
- GroupNames: C₅Character table: identity column and real/complex/symplectic annotationsOpen source
- GAP · Characters and symmetrizationsIrr and section 71.12-5 IndicatorOpen source
group:c-5#representations-2Structure and conventions
For a generator a, χₖ(a) = exp(2πik/5); source row labels need not be ordered by k.
- GroupNames: C₅Character tableOpen source
group:c-5#structure-1