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Questions and Answers
Which property defines the associative property of a group (G, *)?
Which property defines the associative property of a group (G, *)?
- There exists e ∈ G such that for all a ∈ G, a * e = a and e * a = a.
- For all a, b, c ∈ G, a * (b * c) = (a * b) * c. (correct)
- For all a ∈ G, there exists a⁻¹ ∈ G such that a * a⁻¹ = e and a⁻¹ * a = e.
- G is a non-empty set.
What is the role of the element 'e' in the group definition?
What is the role of the element 'e' in the group definition?
- It serves as the identity element. (correct)
- It is an arbitrary element.
- It represents the inverse of every element.
- It demonstrates the closure property.
Which of the following expressions represents the inverse property in a group (G, *)?
Which of the following expressions represents the inverse property in a group (G, *)?
- a * b = b * a
- a * a⁻¹ = e (correct)
- a * (b * c) = (a * b) * c
- a * e = a
In the context of group theory, what does the notation (G, *) signify?
In the context of group theory, what does the notation (G, *) signify?
Which of the following is NOT a required property for (G, *) to be considered a group?
Which of the following is NOT a required property for (G, *) to be considered a group?
If 'a' is an element of group G, what does a⁻¹ represent?
If 'a' is an element of group G, what does a⁻¹ represent?
Given a group (G, *) and elements a, b ∈ G, which expression is guaranteed to be true based solely on the group axioms?
Given a group (G, *) and elements a, b ∈ G, which expression is guaranteed to be true based solely on the group axioms?
What condition must be met for a set G with a binary operation * to satisfy the identity element property?
What condition must be met for a set G with a binary operation * to satisfy the identity element property?
If (G, *) is a group and a ∈ G, which of the following guarantees the existence of an element that, when combined with 'a' using *, results in the identity element 'e'?
If (G, *) is a group and a ∈ G, which of the following guarantees the existence of an element that, when combined with 'a' using *, results in the identity element 'e'?
Which of the following best describes the significance of the binary operation '*' in the context of a group (G, *)?
Which of the following best describes the significance of the binary operation '*' in the context of a group (G, *)?
Flashcards
What is a Group?
What is a Group?
A group consists of a nonempty set G and a binary operation ∗ on G, satisfying associativity, the existence of an identity element, and the existence of inverse elements for each element in G.
Associativity (G1)
Associativity (G1)
For all a, b, c in G, a ∗ (b ∗ c) = (a ∗ b) ∗ c. This means the order in which you perform the operation doesn't change the result.
Identity Element (G2)
Identity Element (G2)
There exists an element e in G such that for all a in G, a ∗ e = a and e ∗ a = a. This special element doesn't change any element when combined.
Inverse Element (G3)
Inverse Element (G3)
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Study Notes
- A group consists of a nonempty set G and a binary operation ∗.
- The group must satisfy three properties: associativity, existence of an identity element, and existence of inverse elements.
Associativity (G1)
- For all elements a, b, and c in G, the equation a ∗ (b ∗ c) = (a ∗ b) ∗ c holds, ensuring the order of operations does not affect the result.
Identity Element (G2)
- There exists an element e in G such that for any element a in G, a ∗ e = a and e ∗ a = a, where e is the identity element.
Inverse Element (G3)
- For every element a in G, there exists an element a^(-1) in G such that a ∗ a^(-1) = e and a^(-1) ∗ a = e, where a^(-1) is the inverse of a.
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