Find all subgroups of order 4 for the symmetric groupS 4 . What I know so far, T

Question

Find all subgroups of order 4 for the symmetric groupS4 . What I know so far, The symmetric group S4 is all permuations of theset A={1,2,3,4} I know that each subgroup of S4 must contain theidentity (1 2 3 4). Since I am looking for subgroups of order4, I am looking for groups with 3 remaining elements that arepermutations of A, each having inverses and are closed undercomposition. How do I find these? Any help is greatly appreciated. Promise to rate.Thanks. Find all subgroups of order 4 for the symmetric groupS4 . What I know so far, The symmetric group S4 is all permuations of theset A={1,2,3,4} I know that each subgroup of S4 must contain theidentity (1 2 3 4). Since I am looking for subgroups of order4, I am looking for groups with 3 remaining elements that arepermutations of A, each having inverses and are closed undercomposition. How do I find these? Any help is greatly appreciated. Promise to rate.Thanks. . What I know so far, The symmetric group S4 is all permuations of theset A={1,2,3,4} I know that each subgroup of S4 must contain theidentity (1 2 3 4). Since I am looking for subgroups of order4, I am looking for groups with 3 remaining elements that arepermutations of A, each having inverses and are closed undercomposition. How do I find these? Any help is greatly appreciated. Promise to rate.Thanks.

Explanation / Answer

The identity is not (1 2 3 4). It is (1). (1 2 3 4) infact generates a subgroup of order 4:{(1),(1,2,3,4),(1,3)(2,4),(1,4,3,2)}). Other cyclic subgroups oforder 4 are {(1),(1,2,4,3),(1,4)(3,2),(1,3,4,2)}, and{(1),(1,3,2,4),(1,2)(3,4),(1,4,2,3)}. So there are 3 cyclicsubgroups of order 4 in S4. There are 3 non-normal subgroups generated by two disjointtranspositions of the type called Klein's 4 group:{(1),(1,2),(3,4),(1,2)(3,4)}, {(1),(1,3),(2,4),(1,3)(2,4)},{(1),(1,4),(2,3),(1,4)(2,3)}. There is one more subgroup comprising the identity and thethree double transpositions: {(1),(1,2)(3,4),(1,3)(2,4),(1,4)(2,3)}. This is alsoisomorphic to Klein's 4 group. There are 3 non-normal subgroups generated by two disjointtranspositions of the type called Klein's 4 group:{(1),(1,2),(3,4),(1,2)(3,4)}, {(1),(1,3),(2,4),(1,3)(2,4)},{(1),(1,4),(2,3),(1,4)(2,3)}. There is one more subgroup comprising the identity and thethree double transpositions: {(1),(1,2)(3,4),(1,3)(2,4),(1,4)(2,3)}. This is alsoisomorphic to Klein's 4 group.

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