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Math Help - Find all the distinct cyclic subgroup of A4

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    Find all the distinct cyclic subgroup of A4

    Find all the distinct cyclic subgroup of A4

    what is distinct cyclic subgroup? Does it mean A4 = {1,2,3,4} ?
    to find all of the subgroup, should it be: (1,2,3,4) , (2,1,3,4) , ( 1,2,4,3) etc ... or it should be (1,2) , (1,3), (1,4) ???
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by rainyice View Post
    Find all the distinct cyclic subgroup of A4

    what is distinct cyclic subgroup? Does it mean A4 = {1,2,3,4} ?
    to find all of the subgroup, should it be: (1,2,3,4) , (2,1,3,4) , ( 1,2,4,3) etc ... or it should be (1,2) , (1,3), (1,4) ???
    I hope you know that A_4=\eta^{-1}(1) where \eta:S_4\to \{-1,1\}:\sigma\mapsto\text{sgn }\sigma. And then a cyclic subgroup of A_4 would be a subgroup that is generated by one element. What's the problem?
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    Still confused

    since you said "then a cyclic subgroup of would be a subgroup that is generated by one element." Would that mean the answer is simply {(1), (2), (3), (4)} there are 4 distinct cyclic subgroups?
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by natmov85 View Post
    since you said "then a cyclic subgroup of would be a subgroup that is generated by one element." Would that mean the answer is simply {(1), (2), (3), (4)} there are 4 distinct cyclic subgroups?
    Whose to say those are all distinct? I mean you could cover all of the cyclic subgroups of G in the class \left\{\langle g\rangle:g\in G\right\} but you wouldn't know which are distinct.
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    Quote Originally Posted by natmov85 View Post
    since you said "then a cyclic subgroup of would be a subgroup that is generated by one element." Would that mean the answer is simply {(1), (2), (3), (4)} there are 4 distinct cyclic subgroups?

    List the different order out, and then find the cyclic subgroup. If you see any of them have the same cyclic subgroup, then just take one of them.
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