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Questions and Answers
What is the identity element in a group?
What is the identity element in a group?
- An element that can never be combined with itself
- An element that does not affect other elements during operation (correct)
- An element that only exists in finite groups
- An element that can be expressed as a product of other elements
Which of the following properties must a set have to be considered a group?
Which of the following properties must a set have to be considered a group?
- Closure, inverses, and associativity (correct)
- Closure, identity, and commutativity
- Closure, inverses, and order
- Associativity, identity, and finiteness
What does it mean for a group to be abelian?
What does it mean for a group to be abelian?
- Each element has a unique inverse in the group
- The group operation is commutative for all elements (correct)
- Elements can only be combined in pairs
- The group is finite and has a limited number of elements
Which of the following is not a type of subgroup?
Which of the following is not a type of subgroup?
What is the order of a group?
What is the order of a group?
Flashcards
Identity Element
Identity Element
An element in a group that leaves other elements unchanged when combined using the group operation.
Group Properties
Group Properties
A set with a binary operation satisfying closure, associativity, identity, and inverse properties.
Abelian Group
Abelian Group
A group where the order of elements in the operation doesn't matter. The operation is commutative.
Order of a Group
Order of a Group
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Subgroup
Subgroup
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Study Notes
The Identity Element
- The identity element in a group is an element that, when combined with any other element in the group using the group's operation, results in that same element.
- The identity element acts like a neutral element in the group operation.
Properties of a Group
- Closure: The result of combining any two elements in the group using the group's operation must also be an element within the group.
- Associativity: The way elements are grouped when combined using the group's operation doesn't affect the final result.
- Identity: The group must contain an identity element.
- Inverse: For every element in the group, there must be an inverse element such that combining them using the group's operation results in the identity element.
Abelian Group
- An Abelian group is a group whose operation is commutative. This means that the order in which elements are combined doesn't affect the result.
Types of Subgroups
- Normal Subgroup: A subgroup where the left and right cosets are equal for every element in the group.
- Cyclic Subgroup: A subgroup generated by a single element.
- Proper Subgroup: A subgroup strictly smaller than the original group.
- Trivial Subgroup: A subgroup containing only the identity element.
Order of a Group
- The order of a group is the number of elements it contains.
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