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    Abstract Algebra: Symmetric Group Problem

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    a) Let G = S_4. What orders do the elements have? Give reasons and examples.

    b) Without listing them, how many subgroups does G have of order 3? Why?

    c) Using examples and/or theorems, argue that G has at least one subgroup of every order dividing |G|.

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    Solution Preview

    a) Since G = S_4, then each element can be expressed as multiplication of disjoint cycles. Elements of S_4 can
    only have 2-cycle, 3-cycle and 4-cycle. So each element can be a 2-cycle, 3-cycle, 4-cycle ...

    Solution Summary

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