# Modern/Abstract Algebra Problem (Group Theory)

What are the normal subgroups in S_4?

I would appreciate it if someone could also give a thourough explanation of how they came up with the answer.

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#### Solution Preview

Please see the attached document.

Solution. Set of permutations {1,2,3,4} with product operation of composition forms a group. Order of S4 is n*(n-1)*...*2.1=24 elements.

Permutation types of {1,2,3,4} are

Four 1 cycles (1)(2)(3)(4) identity 1234ïƒ¨1234 4!/(1^4*4!)=1 g1

One 2 cycle (12)(3)(4) transposition ...

#### Solution Summary

Normal subgroups are determined for modern/abstract ground theory. .

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