# Definition:Linear Group Action

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## Definition

Let $\struct {V, +, \cdot}$ be a vector space over a field $\struct {k, \oplus, \circ}$.

Let $G$ be a group.

Let $\phi : G \times V \to V$ be an action of $G$ on $V$.

Then $\phi$ is a **(left) linear group action** if and only if it is compatible with the linear structure of $V$ in the following sense:

- $(1): \quad \forall v_1, v_2 \in V: g \in G: \map \phi {g, v_1 + v_2} = \map \phi {g, v_1} + \map \phi {g, v_2}$

- $(2): \quad \forall \lambda \in k, g \in G, v \in V: \map \phi {g, \lambda \cdot v} = \lambda \cdot \map \phi {g, v}$

### Right Linear Group Action

Definition:Linear Group Action/Right Linear Group Action

## Also see

- Definition:Linear Representation
- Correspondence between Linear Group Actions and Linear Representations

## Sources

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