15.2.4 - Generating Cyclic Subgroups
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Practice Questions
Test your understanding with targeted questions
Define what makes a subgroup.
💡 Hint: Recall the group axioms.
What is the closure property?
💡 Hint: Think of operations in a group.
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Interactive Quizzes
Quick quizzes to reinforce your learning
What is the definition of a subgroup?
💡 Hint: Consider the properties related to subsets.
Can closure alone guarantee a subset is a subgroup in infinite groups?
💡 Hint: Think about the implications of it being infinite.
1 more question available
Challenge Problems
Push your limits with advanced challenges
Prove that every cyclic subgroup is abelian.
💡 Hint: Think of how elements generated by powers commute.
Given the group of real numbers under addition, show that subsets like {x ∈ ℝ | x ≤ 0} and {x ∈ ℝ | x ≥ 0} are not subgroups.
💡 Hint: Can you find elements in the subsets that fail closure?
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