# Modular representation theory of cyclic group:Z3

This article gives specific information, namely, modular representation theory, about a particular group, namely: cyclic group:Z3.

View modular representation theory of particular groups | View other specific information about cyclic group:Z3

This article describes the modular representation theory of cyclic group:Z3, the cyclic group of order three, i.e., the linear representation theory of the group in characteristic three, which means the linear representation theory over field:F3 and its extensions.

For the linear representation theory in other characteristics, see linear representation theory of cyclic group:Z3.

## Summary

Item | Value |
---|---|

degrees of irreducible representations (or equivalently, degrees of irreducible Brauer characters) | 1 maximum: 1, lcm: 1, number: 1 |

smallest field of realization of irreducible representations in characteristic 3 | field:F3, i.e., the field of three elements |

degrees of indecomposable representations | 1,2,3 maximum: 3 |

smallest field of realization of indecomposable representations in characteristic 3 | field:F3, i.e., the field of three elements |

## Irreducible representations

`Further information: Irreducible representation of group of prime power order in characteristic equal to underlying prime is trivial`

There is a *unique* irreducible representation: the trivial representation, which sends all elements of the group to the matrix . This is a *general* feature common to *all* representations of a group of prime power order in a field of characteristic equal to the prime.

## Indecomposable representations

There are (up to equivalence of linear representations) three indecomposable representations:

Representation | Degree | Image of identity element | Image of generator of group | Image of other generator of group |
---|---|---|---|---|

trivial | 1 | |||

two-dimensional indecomposable | 2 | |||

three-dimensional indecomposable | 3 |