Practice Order Of A Group Element (14.4) - Cyclic Groups - Discrete Mathematics

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6 sample questions

Question 1.
What is the identity element in a group?
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Think about the role of 0 in addition or 1 in multiplication.
Question 2.
Define the order of an element.
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What do you get when you finally reach the group's starting point?
3Question 3.
What is the order of an element?
- The smallest positive integer for multiplication
- The highest exponent possible
- A measure of the group size
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Think about the cycle and repetition in operations.
4Question 4.
In a cyclic group, can you have multiple generators?
- True
- False
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Consider the number of ways you can reach a full cycle.
Question 5.
Prove that if a group has an element with infinite order, it cannot be finite.
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Consider the implications of having a finite number of elements but extending indefinitely with one.
Question 6.
Given a group (Z/7Z, +), describe its properties and the order of each element.
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Think about how addition mod 7 would work, and apply the elementary properties.

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