# Linearly primitive group

From Groupprops

This article defines a property that can be evaluated for finite groups (and hence, a particular kind of group property)

View other properties of finite groups OR View all group properties

## Contents

## Definition

### Symbol-free definition

A finite group is said to be **linearly primitive** if it has a faithful irreducible representation over an algebraically closed field of characteristic zero (or equivalently, over the algebraic closure of rationals or over the complex numbers).

### Definition with symbols

A group is said to be linearly primitive if there is a homomorphism for some vector space over the complex numbers, such that has no proper nonzero -invariant subspace.

## Relation with other properties

### Stronger properties

### Weaker properties

- Cyclic-center group:
`For full proof, refer: Linearly primitive implies cyclic-center`