Finite groups · Group

$\mathrm{C}_{3}$ Cyclic group of order three

The group generated by a single element r with r³ = 1. All ordinary complex irreducible representations are one-dimensional.

Compare this group

Executable coverageDiscrete charges

Definition

⟨r | r³ = 1⟩

Definition sources

Order: $3$
Center: $\mathrm{C}_{3}$

Reference properties

Property	Value	Evidence
Order	$3$	Sources
Center	$\mathrm{C}_{3}$	Sources
Abelian	Yes	Sources
Simple finite group	Yes	Sources
Ordinary complex irrep dimensions	1, 1, 1	Sources

Representations

The generator acts by 1, ω, or ω², with ω = exp(2πi/3). Complex conjugation exchanges χ and χ². [sources]

Complete ordinary complex irrep dimension and reality summary
Irrep label	Complex dimension	Reality type
$1$	1	Real
χ	1	Complex
χ²	1	Complex

Complete dimension and reality summary for ordinary complex irreps. This does not supply explicit matrices, every character value, or modular representations. Labels follow the conventions stated in this record.

In physics

Discrete charge assignments and elementary selection rules. [sources]

Names and aliases

Z3 · ℤ₃

Aliases identify the stated object under this record’s conventions. Type names and broader search terms are kept separate.

Related objects and maps

$\mathrm{A}_{4}$
Alternating group on four letters
Related group
$\mathrm{SU}(3)$
Special unitary group
Related group
$\mathrm{C}_{n}$
Finite cyclic groups
Cyclic family
$\mathrm{C}_{p}$
Cyclic groups of prime order
Prime-order simple 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

⟨r | r³ = 1⟩