Practice Generating Cyclic Subgroups (15.2.4) - Subgroups - Discrete Mathematics

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6 sample questions

Question 1.
Define what makes a subgroup.
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Recall the group axioms.
Question 2.
What is the closure property?
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Think of operations in a group.
3Question 3.
What is the definition of a subgroup?
- A subgroup can only be empty
- A non-empty subset that is a group under the same operation
- Any subset of a group
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Consider the properties related to subsets.
4Question 4.
Can closure alone guarantee a subset is a subgroup in infinite groups?
- True
- False
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Think about the implications of it being infinite.
Question 5.
Prove that every cyclic subgroup is abelian.
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Think of how elements generated by powers commute.
Question 6.
Given the group of real numbers under addition, show that subsets like {x ∈ ℝ | x ≤ 0} and {x ∈ ℝ | x ≥ 0} are not subgroups.
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Can you find elements in the subsets that fail closure?

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