Practice Subgroups (15.2) - Subgroups - Discrete Mathematics

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8 questions

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6 sample questions

Question 1.
Define a subgroup in your own words.
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Think of it as a smaller group that still behaves like the big group.
Question 2.
What are the properties needed for a subset to be a subgroup?
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There are three main points to remember.
3Question 3.
Is a non-empty subset of a group always a subgroup?
- True
- False
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Think about the defining properties of subgroups.
4Question 4.
What must be true for any two elements in a subgroup?
- Their sum must be in the subgroup
- One of them must be the identity
- Their product must be zero
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Remember the definition of closure!
Question 5.
Given the group Z_12, identify all possible subgroups and their generators.
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Consider how elements combine under addition mod 12.
Question 6.
Prove that every subgroup of a cyclic group is cyclic.
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Reflect on what it means for a generator to create elements of the group.

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