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Cyclic Groups and Subgroup Lattices
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Cyclic Groups and Subgroup Lattices

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Questions and Answers

What is the defining characteristic of a cyclic group?

  • Has an infinite number of elements
  • Contains elements that are not powers of the generator
  • Generated by multiple elements
  • Generated by a single element (correct)

    • If a group G can be expressed as G=g, what does this notation signify?

  • G is not cyclic
  • G contains no subgroups
  • G is generated by the element g (correct)
  • G has an infinite order

    • What is the order of a cyclic group defined as?

  • Order the group was created in
  • Number of elements in the group (correct)
  • Number of times the generator has been multiplied by itself
  • Number of distinct generators in the group

    • If a group can be generated by one element but contains other elements not expressible as powers of the generator, is it cyclic?

    <p>No</p> Signup and view all the answers

    In a cyclic group, what can be said about every subgroup?

    <p>It is cyclic</p> Signup and view all the answers

    Are all subgroups of a cyclic group unique in their order?

    <p>Yes</p> Signup and view all the answers

    How do isomorphism and cyclic groups relate?

    <p>Cyclic groups of different orders are isomorphic</p> Signup and view all the answers

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