Alternating Group A_n - What Is Subgroup with Index n?

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SUMMARY

The alternating group A_n cannot have subgroups with an index less than n, as established in group theory. The subgroup with an index equal to n is A_{n-1}, which is isomorphic to several subgroups of A_n, each corresponding to fixing a different element from the set {1, 2, ..., n}. This highlights the structural properties of A_n and its relationship with its subgroups.

PREREQUISITES
  • Understanding of group theory concepts, specifically alternating groups.
  • Familiarity with subgroup index and isomorphism.
  • Knowledge of the notation and properties of A_n and A_{n-1}.
  • Basic comprehension of permutation groups.
NEXT STEPS
  • Research the properties of alternating groups A_n and their subgroups.
  • Study the concept of subgroup index in group theory.
  • Explore the isomorphism between groups, particularly in the context of A_n and A_{n-1}.
  • Learn about the action of groups on sets and how it relates to subgroup structures.
USEFUL FOR

Mathematicians, students of abstract algebra, and anyone studying group theory, particularly those interested in the properties of alternating groups and their substructures.

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I have seen proofs that the alternating group A_n cannot have subgroups with index less than n. Ok, but what is the subgroup with index equal to n?
 
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A_{n-1}?
 
morphism said:
A_{n-1}?

this is true, but one should point out there there are usually several subgroups of An isomorphic to An-1 (each one fixing a different element of {1,2,...,n}).
 

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