Showing Subgroups of a Permutation Group are Isomorphic

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SUMMARY

The discussion centers on determining the isomorphism between two subgroups of the symmetric group S6, specifically G=(123), (123)(456) and H=(14), (123)(456). Both groups have the same cardinality, but the validity of H as a subgroup is questioned due to the element (14)(123)(456) resulting in (123456), which is not contained in H. The correct definitions of the groups are crucial for establishing their isomorphism.

PREREQUISITES
  • Understanding of group theory concepts, specifically subgroups and isomorphism.
  • Familiarity with symmetric groups, particularly S6.
  • Knowledge of permutation notation and operations.
  • Basic skills in abstract algebra.
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  • Study the properties of symmetric groups, focusing on S6.
  • Learn about subgroup criteria and how to verify subgroup status.
  • Explore the concept of isomorphism in group theory.
  • Investigate examples of isomorphic and non-isomorphic groups.
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Mathematicians, students of abstract algebra, and anyone interested in group theory and its applications in permutations.

Obraz35
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Define two subgroups of S6:
G=[e, (123), (123)(456)]
H=[e, (14), (123)(456)]

Determine whether G and H are isomorphic.

It seems as if they should be since they have the same cardinality and you can certainly map the elements to one another, but I don't know what other factors need to be considered when deciding whether they are isomorphic.
 
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H is not a subgroup since (14)(123)(456) = (123456) which is not in H. Are you sure you've written down the correct group?
 
Oops. I meant G=<(123), (123)(456)> and H=<(14), (123)(456)>.
 

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