# Adjoint group structures for cyclic group:Z4

From Groupprops

## Contents |

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

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

There exist two ways of making cyclic group:Z4 into the adjoint group of a radical ring. One of these makes it an algebra group. The other simply uses a ring with the same additive group.

## Summary

Radical ring | Additive group | Simplest commutative unital rings over which it is an algebra | Maximum of element nilpotencies | Nilpotency index of whole ring |
---|---|---|---|---|

nilpotent over field:F2, where one basis element squares to the other and all basis products are zero (see algebra group structures for cyclic group:Z4) | Klein four-group | field:F2 | 3 | 3 |

has additive group cyclic group:Z4, multiplication is identically zero | cyclic group:Z4 | ring:Z4 | 2 | 2 |