Linearly primitive group

This article defines a property that can be evaluated for finite groups (and hence, a particular kind of group property)
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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 $G$ is said to be linearly primitive if there is a homomorphism $\sigma:G \to GL(V)$ for some vector space $V$ over the complex numbers, such that $V$ has no proper nonzero $G$-invariant subspace.

Relation with other properties

Stronger properties

• Finite simple group

Weaker properties

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