Category:Group Actions
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This category contains results about Group Actions.
Definitions specific to this category can be found in Definitions/Group Actions.
Let $X$ be a set.
Let $\struct {G, \circ}$ be a group whose identity is $e$.
Left Group Action
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:
| \((\text {GA} 1)\) | $:$ | \(\ds \forall g, h \in G, x \in X:\) | \(\ds g * \paren {h * x} = \paren {g \circ h} * x \) | ||||||
| \((\text {GA} 2)\) | $:$ | \(\ds \forall x \in X:\) | \(\ds e * x = x \) |
Right Group Action
A right group action is a mapping $\phi: X \times G \to X$ such that:
- $\forall \tuple {x, g} \in X \times G : x * g := \map \phi {x, g} \in X$
in such a way that the right group action axioms are satisfied:
| \((\text {RGA} 1)\) | $:$ | \(\ds \forall g, h \in G, x \in X:\) | \(\ds \paren {x * g} * h = x * \paren {g \circ h} \) | ||||||
| \((\text {RGA} 2)\) | $:$ | \(\ds \forall x \in X:\) | \(\ds x * e = x \) |
Subcategories
This category has the following 8 subcategories, out of 8 total.
A
- Automorphic Functions (Group Theory) (empty)
E
F
- Faithful Group Actions (empty)
O
- Orbit-Stabilizer Theorem (4 P)
S
- Stabilizer is Subgroup (2 P)
T
- Topological Group Actions (2 P)
- Transitive Group Actions (7 P)
Pages in category "Group Actions"
The following 32 pages are in this category, out of 32 total.
C
G
- Group Action defines Permutation Representation
- Group Action determines Bijection
- Group Action Induces Equivalence Relation
- Group Action on Subgroup by Left Regular Representation
- Group Action on Subgroup by Right Regular Representation
- Group Action on Subgroup of Symmetric Group
- Group Acts by Homeomorphisms Implies Projection on Quotient Space is Open
- Group Acts on Itself