- #1
dimuk
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I need a proof of any subgroup of S_n which is isomorphic to S_(n-1) fixes a point in {1, 2,..., n} unless n=6.
This refers to a subgroup of the symmetric group on n elements (Sym(n)) that is isomorphic (structurally equivalent) to the symmetric group on (n-1) elements (S_(n-1)), with the exception of when n=6.
This subgroup is important because it has unique properties that make it different from other subgroups of Sym(n). In particular, it has a special relationship with the symmetric group on (n-1) elements, making it a useful tool in understanding and studying group theory.
The subgroup is constructed by taking the elements of Sym(n) that fix the last element (n) and removing the element that swaps the first and last elements, which creates a subgroup isomorphic to S_(n-1).
This subgroup reveals that there is a unique relationship between Sym(n) and S_(n-1), as seen by their isomorphism. It also shows that the symmetric group on 6 elements (Sym(6)) is different from other symmetric groups.
While this subgroup may not have direct applications in the real world, it is a fundamental concept in group theory and has implications for understanding the structure and properties of groups. This can have applications in fields such as physics, cryptography, and computer science.