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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?
morphism said:A_{n-1}?
The alternating group An is a subgroup of the symmetric group Sn consisting of all even permutations of n distinct objects. In other words, it is the group of all permutations that can be obtained by an even number of transpositions.
A subgroup is a subset of a group that itself forms a group under the same operation as the original group. In other words, it is a smaller group that shares the same structure and properties as the larger group.
The index of a subgroup is the number of cosets (distinct left or right cosets) of the subgroup in the larger group. It is denoted by [G : H], where G is the larger group and H is the subgroup. In the case of the alternating group An, the index is equal to n!/2.
Subgroups with index n in the alternating group An are important because they represent the symmetries of regular n-gons. This means that they are the groups of transformations that preserve the shape and size of a regular n-sided polygon.
Yes, there are several practical applications of subgroups with index n in the alternating group An. For example, they are used in the study of crystallography to describe the symmetries of crystals, as well as in the field of group theory to study the structure and properties of groups. They also have applications in coding theory and cryptography.