# Algebra group structures for cyclic group:Z4

From Groupprops

This article gives specific information, namely, algebra group structures, about a particular group, namely: cyclic group:Z4.

View algebra group structures for particular groups | View other specific information about cyclic group:Z4

There exists a unique way of making cyclic group:Z4 into an algebra group over field:F2. It is not an algebra group over any other field. This is unique in a strict sense, not merely up to isomorphisms.

## Summary

Algebra | Smallest for which it can be embedded in | Maximum of element nilpotencies | Nilpotency index of whole algebra |
---|---|---|---|

the only possible one, where one generator squares to the other | 3 | 3 | 3 |

## Algebra structure

### Multiplication table (structure constants)

The algebra is two-dimensional. It has the following multiplication table, with basis elements . Note that Adjoint group of a radical ring is abelian iff the radical ring is commutative, so the algebra group is commutative, and we don't have to worry about the order of multiplication:

0 | ||

0 | 0 |

### Verification of properties

- is associative: All products of length three or more involving basis elements are zero, hence, by linearity, all products of length three or more are zero.
- is nilpotent: All products of length three or more involving basis elements are zero, hence, by linearity, all products of length three or more are zero.
- The algebra group for is cyclic of order 4: Indeed, and , so is an element of order four (see also powering map by field characteristic is same in algebra and algebra group).

### Description as subalgebra of niltriangular matrix Lie algebra

The algebra can be realized explicitly as a subalgebra of niltriangular matrix Lie algebra:NT(3,2) as follows: