Group Actions and Symmetry
| Description: This quiz covers the fundamental concepts and applications of group actions and symmetry in mathematics. | |
| Number of Questions: 15 | |
| Created by: Aliensbrain Bot | |
| Tags: group actions symmetry group theory transformation groups |
What is a group action?
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A mapping from a group to a set
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A transformation of a set by a group
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A group of transformations of a set
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A set of transformations that preserve a given structure
A group action is a mapping from a group to a set of transformations of another set, such that the composition of two transformations in the group corresponds to the composition of the corresponding transformations in the set.
What is the orbit of an element under a group action?
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The set of all elements that are transformed into the given element
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The set of all elements that transform the given element into itself
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The set of all elements that commute with the given element
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The set of all elements that are fixed by the given element
The orbit of an element under a group action is the set of all elements that are transformed into the given element by the action of the group.
What is the stabilizer of an element under a group action?
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The set of all elements that are transformed into the given element
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The set of all elements that transform the given element into itself
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The set of all elements that commute with the given element
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The set of all elements that are fixed by the given element
The stabilizer of an element under a group action is the set of all elements in the group that transform the given element into itself.
What is the kernel of a group action?
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The set of all elements that are transformed into the identity element
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The set of all elements that transform the identity element into itself
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The set of all elements that commute with the identity element
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The set of all elements that are fixed by the identity element
The kernel of a group action is the set of all elements in the group that transform the identity element into itself.
What is the fixed point set of a group action?
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The set of all elements that are transformed into themselves
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The set of all elements that transform themselves into themselves
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The set of all elements that commute with themselves
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The set of all elements that are fixed by themselves
The fixed point set of a group action is the set of all elements in the set that are transformed into themselves by the action of the group.
What is the isotropy subgroup of an element under a group action?
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The set of all elements that are transformed into the given element
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The set of all elements that transform the given element into itself
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The set of all elements that commute with the given element
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The set of all elements that are fixed by the given element
The isotropy subgroup of an element under a group action is the set of all elements in the group that transform the given element into itself.
What is the orbit-stabilizer theorem?
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The order of the orbit of an element is equal to the index of the stabilizer of the element
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The order of the stabilizer of an element is equal to the index of the orbit of the element
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The order of the orbit of an element is equal to the order of the stabilizer of the element
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The order of the stabilizer of an element is equal to the order of the orbit of the element
The orbit-stabilizer theorem states that the order of the orbit of an element under a group action is equal to the index of the stabilizer of the element in the group.
What is a transitive group action?
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A group action in which every element of the set is transformed into every other element
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A group action in which every element of the set is transformed into itself
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A group action in which every element of the set is transformed into a unique element
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A group action in which every element of the set is transformed into a finite number of elements
A transitive group action is a group action in which every element of the set is transformed into every other element by the action of the group.
What is a regular group action?
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A transitive group action in which the stabilizer of every element is trivial
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A transitive group action in which the stabilizer of every element is finite
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A transitive group action in which the stabilizer of every element is infinite
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A transitive group action in which the stabilizer of every element is non-trivial
A regular group action is a transitive group action in which the stabilizer of every element is trivial.
What is a permutation group?
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A group of transformations of a set
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A group of transformations of a set that preserves a given structure
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A group of transformations of a set that is transitive
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A group of transformations of a set that is regular
A permutation group is a group of transformations of a set.
What is a symmetric group?
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A permutation group that is transitive
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A permutation group that is regular
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A permutation group that is both transitive and regular
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A permutation group that is neither transitive nor regular
A symmetric group is a permutation group that is both transitive and regular.
What is the alternating group?
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The subgroup of the symmetric group consisting of all even permutations
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The subgroup of the symmetric group consisting of all odd permutations
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The subgroup of the symmetric group consisting of all permutations that fix a given element
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The subgroup of the symmetric group consisting of all permutations that do not fix any element
The alternating group is the subgroup of the symmetric group consisting of all even permutations.
What is the dihedral group?
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The group of symmetries of a regular polygon
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The group of symmetries of a regular polyhedron
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The group of symmetries of a sphere
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The group of symmetries of a cube
The dihedral group is the group of symmetries of a regular polygon.
What is the octahedral group?
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The group of symmetries of a cube
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The group of symmetries of a regular octahedron
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The group of symmetries of a regular dodecahedron
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The group of symmetries of a regular icosahedron
The octahedral group is the group of symmetries of a regular octahedron.
What is the icosahedral group?
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The group of symmetries of a cube
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The group of symmetries of a regular octahedron
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The group of symmetries of a regular dodecahedron
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The group of symmetries of a regular icosahedron
The icosahedral group is the group of symmetries of a regular icosahedron.