Practice Properties Of Order Of A Group Element (14.5) - Cyclic Groups

Practice: Properties of Order of a Group Element

Lesson 14.5Cyclic Groups

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10 questions

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6 sample questions

Question 1.
Define the order of an element in a group. Give an example.
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Think about how many times you need to apply the operation to get back to the identity.
Question 2.
What is the identity element in a group?
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Consider the element that allows you to act without changing value.
3Question 3.
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
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Think about what it means for an element to return to the starting point.
4Question 4.
True or False: An element with infinite order will eventually repeat itself.
- True
- False
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Consider the definition of infinite versus finite orders.
Question 5.
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.
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Think about how many distinct elements can be produced by the powers of each element.
Question 6.
Demonstrate the necessity of the condition that the order of a generator must match the order of the cyclic group it generates.
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Reflect on what it means to cycle back through generated elements.

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