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Questions and Answers
What is the identity element in the given group G with respect to the operation ?
What is the identity element in the given group G with respect to the operation ?
- 6
- 8 (correct)
- 4
- 2
Which of the following demonstrates closure for the operation in group G?
Which of the following demonstrates closure for the operation in group G?
- The operation always produces an identity element.
- All elements of the Cayley table are members of the set S. (correct)
- Multiplying any two elements results in an element that is not in G.
- Every element has a unique representation in the table.
What is the inverse of the element 6 in group G?
What is the inverse of the element 6 in group G?
- 4
- 10 (correct)
- 12
- 2
Which property of the operation * is confirmed by the statement 'Multiplication modulo n is associative'?
Which property of the operation * is confirmed by the statement 'Multiplication modulo n is associative'?
Which pair represents the correct relationship in terms of inverses within group G?
Which pair represents the correct relationship in terms of inverses within group G?
Which property does NOT generally apply to all groups in matrix multiplication?
Which property does NOT generally apply to all groups in matrix multiplication?
What is the unique identity element for multiplication in the set of non-zero rational numbers (Q*)?
What is the unique identity element for multiplication in the set of non-zero rational numbers (Q*)?
What characteristic identifies an Abelian group in its Cayley table?
What characteristic identifies an Abelian group in its Cayley table?
How many inverses does each element have in a group?
How many inverses does each element have in a group?
In the binary operation of multiplication modulo 14, which of the following is an example of an element?
In the binary operation of multiplication modulo 14, which of the following is an example of an element?
What does the inverse of an element combined with that element produce?
What does the inverse of an element combined with that element produce?
Which of the following describes the Commutative Property in an Abelian group?
Which of the following describes the Commutative Property in an Abelian group?
Which of the following is true about identity in groups?
Which of the following is true about identity in groups?
What fundamental property ensures that the combination of any two elements in a group yields another element within the same group?
What fundamental property ensures that the combination of any two elements in a group yields another element within the same group?
Which of the following describes a group that is not Abelian?
Which of the following describes a group that is not Abelian?
Which binary operation is an example of a non-closure scenario in group theory?
Which binary operation is an example of a non-closure scenario in group theory?
In group theory, what does the identity element guarantee?
In group theory, what does the identity element guarantee?
Which group is classified as finite and Abelian?
Which group is classified as finite and Abelian?
What is the associative property in group theory?
What is the associative property in group theory?
Which group can be classified as an infinite Abelian group?
Which group can be classified as an infinite Abelian group?
What does the inverse property guarantee in a group?
What does the inverse property guarantee in a group?
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Study Notes
Group Theory
- Group Theory is a branch of Abstract Algebra that explores the properties of groups.
- A Group (G) is a set of elements with a binary operation (*) that satisfies four properties.
Properties of a Group
- Closure: The result of combining any two elements in the group using the operation is always another element in the group.
- Associativity: The order in which operations are performed does not affect the result.
- Identity: There exists an identity element (e) in the group such that for any element a in G, ae = ea = a. The identity element is unique.
- Inverses: For every element a in G, there exists an inverse element (a⁻¹) such that aa⁻¹ = a⁻¹a = e. Inverses are unique, meaning each element has only one inverse.
Types of Groups
- Groups can be Finite (contain a finite number of elements) or Infinite (contain an infinite number of elements).
- Groups can also be Commutative (also known as Abelian) or Non-Commutative.
- Abelian groups satisfy the commutative property: ab = ba for all elements a and b in G.
- Non-Abelian groups do not satisfy the commutative property.
Examples of Groups
-
Infinite, Abelian:
- The integers under addition (Z, +)
- The rational numbers without zero under multiplication (Q*, X)
-
Infinite, Non-Abelian:
- The non-singular n x n matrices under matrix multiplication.
-
Finite, Abelian:
- The Integers Mod n under Modular Addition (Zn, +)
Cayley Tables
- Cayley tables display the results of the group operation for each pair of elements in a group.
- Abelian groups have a diagonal symmetry in their Cayley tables, making it easier to identify them.
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