Group Theory Overview

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Questions and Answers

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?

  • 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?

  • 4
  • 10 (correct)
  • 12
  • 2

Which property of the operation * is confirmed by the statement 'Multiplication modulo n is associative'?

<p>Associativity (A)</p> Signup and view all the answers

Which pair represents the correct relationship in terms of inverses within group G?

<p>The inverse of 4 is 12 (C)</p> Signup and view all the answers

Which property does NOT generally apply to all groups in matrix multiplication?

<p>Commutative Property (A)</p> Signup and view all the answers

What is the unique identity element for multiplication in the set of non-zero rational numbers (Q*)?

<p>1 (D)</p> Signup and view all the answers

What characteristic identifies an Abelian group in its Cayley table?

<p>Diagonal symmetry (C)</p> Signup and view all the answers

How many inverses does each element have in a group?

<p>Exactly one inverse (A)</p> Signup and view all the answers

In the binary operation of multiplication modulo 14, which of the following is an example of an element?

<p>13 (D)</p> Signup and view all the answers

What does the inverse of an element combined with that element produce?

<p>The identity element (D)</p> Signup and view all the answers

Which of the following describes the Commutative Property in an Abelian group?

<p>a * b = b * a (C)</p> Signup and view all the answers

Which of the following is true about identity in groups?

<p>Only one identity element exists in any group (A)</p> Signup and view all the answers

What fundamental property ensures that the combination of any two elements in a group yields another element within the same group?

<p>Closure (B)</p> Signup and view all the answers

Which of the following describes a group that is not Abelian?

<p>The non-singular n x n matrices under matrix multiplication (C)</p> Signup and view all the answers

Which binary operation is an example of a non-closure scenario in group theory?

<p>Addition of odd integers (B)</p> Signup and view all the answers

In group theory, what does the identity element guarantee?

<p>That there is a special element which does not change any element when combined (A)</p> Signup and view all the answers

Which group is classified as finite and Abelian?

<p>The Integers Mod n under Modular Addition (A)</p> Signup and view all the answers

What is the associative property in group theory?

<p>The combination of three or more elements can be grouped in any way (B)</p> Signup and view all the answers

Which group can be classified as an infinite Abelian group?

<p>The Integers under Addition (B)</p> Signup and view all the answers

What does the inverse property guarantee in a group?

<p>For every element there is exactly one other element that when combined yields the identity element (B)</p> Signup and view all the answers

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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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