Isomorphism in Groups with No Proper Subgroups and Absolute Value Greater than 1

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In summary, the conversation discusses how to prove that a group with no proper, nontrivial subgroups and a size greater than 1 must be isomorphic to Z_p for some prime p. It explores the concepts of isomorphism, cyclic groups, and subgroups. The final step is to prove that every group has a cyclic subgroup.
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kathrynag
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Homework Statement



Let G be any group with no proper , nontrivial subgroups and assume abs value(G)>1. Prove that G must be isomorphic to Z_p for some prime p.

Homework Equations





The Attempt at a Solution


I know we have an isomorphism if a group is 1-1, onto, and the homomorphism property holds.
 
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  • #2


What happens if G is cyclic? In this case, if G does not contain p elements, can you find a nontrivial subgroup? What happens if G is not cyclic? In this case, can you find a nontrivial subgroup?
 
  • #3


G is cyclic means G=<a>, where a is a generator.
 
  • #4


Yes, but reason further. If G is cyclic and its number of elements is composite... is there a nontrivial subgroup?
 
  • #5


I know a cyclic group of order n has exactly one subgroup of order m for each positive divisor m of n.
 
  • #6


So can you prove that if G is cyclic with only trivial subgroups, then G is isomorphic to some Zp with p prime?

The next step after that is to prove that every group has a cyclic subgroup
 

1. What is an isomorphic math problem?

An isomorphic math problem is a type of problem in which the underlying structure or relationships between elements remain the same even if the elements are changed or transformed. Essentially, it is a problem that can be solved using the same methods and principles regardless of the specific values or numbers involved.

2. How is isomorphic math different from regular math?

Regular math problems involve specific numbers and values that must be solved using specific operations or formulas. Isomorphic math, on the other hand, focuses on the underlying structure and relationships between elements, allowing for a more abstract and general approach to problem-solving.

3. Can isomorphic math be applied to real-life situations?

Yes, isomorphic math can be applied to real-life situations, particularly in fields such as computer science, physics, and economics. It can help identify patterns and relationships between different variables and provide a more efficient way of solving problems.

4. Are there any specific strategies for solving isomorphic math problems?

There are several strategies that can be used to solve isomorphic math problems, including breaking down the problem into smaller, more manageable parts, using visual aids such as diagrams or graphs, and identifying and utilizing patterns and symmetries within the problem.

5. How can understanding isomorphic math benefit me?

Understanding isomorphic math can improve problem-solving skills and critical thinking abilities. It can also help in making connections between seemingly unrelated concepts and finding more efficient solutions in various fields such as science, technology, and economics.

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