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    Modern/Abstract Algebra Problem (Group Theory)

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    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 1234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. .