Definition:Group Action/Left Group Action
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Definition
Let $X$ be a set.
Let $\struct {G, \circ}$ be a group whose identity is $e$.
A (left) group action is an operation $\phi: G \times X \to X$ such that:
- $\forall \tuple {g, x} \in G \times X: g * x := \map \phi {g, x} \in X$
in such a way that the group action axioms are satisfied:
\((\text {GA} 1)\) | $:$ | \(\ds \forall g, h \in G, x \in X:\) | \(\ds g * \paren {h * x} = \paren {g \circ h} * x \) | ||||||
\((\text {GA} 2)\) | $:$ | \(\ds \forall x \in X:\) | \(\ds e * x = x \) |
Different Approaches
During the course of an exposition in group theory, it is usual to define a group action as a left group action, without introducing the concept of a right group action.
It is apparent during the conventional development of the subject that there is rarely any need to discriminate between the two approaches.
Hence, on $\mathsf{Pr} \infty \mathsf{fWiki}$, we do not in general consider the right group action, and instead present results from the point of view of left group actions alone.
Also see
Sources
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 5.5$. Groups acting on sets
- 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: The Sylow Theorems: $\S 53$
- 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): $\S 3.3$: Group actions and coset decompositions
- 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $10$: The Orbit-Stabiliser Theorem: Definition $10.1$