Practice Properties of Order of a Group Element - 14.5 | 14. Cyclic Groups | Discrete Mathematics - Vol 3
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14.5 - Properties of Order of a Group Element

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Learning

Practice Questions

Test your understanding with targeted questions related to the topic.

Question 1

Easy

Define the order of an element in a group. Give an example.

💡 Hint: Think about how many times you need to apply the operation to get back to the identity.

Question 2

Easy

What is the identity element in a group?

💡 Hint: Consider the element that allows you to act without changing value.

Practice 4 more questions and get performance evaluation

Interactive Quizzes

Engage in quick quizzes to reinforce what you've learned and check your comprehension.

Question 1

What is the order of an element g in a group?

  • 1. Largest integer such that g^n = g
  • 2. Smallest positive integer n such that g^n = e
  • 3. The total number of elements in the group

💡 Hint: Think about what it means for an element to return to the starting point.

Question 2

True or False: An element with infinite order will eventually repeat itself.

  • True
  • False

💡 Hint: Consider the definition of infinite versus finite orders.

Solve 2 more questions and get performance evaluation

Challenge Problems

Push your limits with challenges.

Question 1

Prove that if in a finite cyclic group the orders of two distinct elements are equal, then those two elements must generate the same subgroup.

💡 Hint: Think about how many distinct elements can be produced by the powers of each element.

Question 2

Demonstrate the necessity of the condition that the order of a generator must match the order of the cyclic group it generates.

💡 Hint: Reflect on what it means to cycle back through generated elements.

Challenge and get performance evaluation