Is subgroup of index 2 in any group is unique?

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I know that subgroup of index 2 is normal .

I am interested in knowing that is that subgroup is unqiue or there exist example of subgroup which can have 2 subgroup of index 2?

If there is example exist , then what is condition on subgroup implies that subgroup of index 2 is unique?

Any Help will be appreciated.






abstract-algebra group-theory normal-subgroups







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    up vote
    2
    down vote

    favorite












    I know that subgroup of index 2 is normal .

    I am interested in knowing that is that subgroup is unqiue or there exist example of subgroup which can have 2 subgroup of index 2?

    If there is example exist , then what is condition on subgroup implies that subgroup of index 2 is unique?

    Any Help will be appreciated.






    abstract-algebra group-theory normal-subgroups







    share|cite|improve this question







    New contributor




    user594188 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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      up vote
      2
      down vote

      favorite









      up vote
      2
      down vote

      favorite











      I know that subgroup of index 2 is normal .

      I am interested in knowing that is that subgroup is unqiue or there exist example of subgroup which can have 2 subgroup of index 2?

      If there is example exist , then what is condition on subgroup implies that subgroup of index 2 is unique?

      Any Help will be appreciated.






      abstract-algebra group-theory normal-subgroups







      share|cite|improve this question







      New contributor




      user594188 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.











      I know that subgroup of index 2 is normal .

      I am interested in knowing that is that subgroup is unqiue or there exist example of subgroup which can have 2 subgroup of index 2?

      If there is example exist , then what is condition on subgroup implies that subgroup of index 2 is unique?

      Any Help will be appreciated.






      abstract-algebra group-theory normal-subgroups




      abstract-algebra group-theory normal-subgroups






      share|cite|improve this question







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      user594188 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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      share|cite|improve this question







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      asked 1 hour ago









      user594188

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          2 Answers
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          Unfortunately it is not even unique up to isomorphism! Take, for instance, $(mathbbZ/2mathbbZ)times (mathbbZ/4mathbbZ)$.






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          • Good example. Another: the dihedral group $D_4$ of order $8$ has at least two subgroups of order $4$: the Klein $4$-group and the subgroup consisting of the rotations. They are also non-iso.
            – Randall
            1 hour ago


















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          No way is it unique (generally). The group $G=mathbbZ_2 times mathbbZ_2$ has three subgroups of index $2$: $mathbbZ_2 times0$, the vice versa, and $(0,0), (1,1)$. You can use the same idea to cook up plenty of other examples.






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            2 Answers
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            active

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            2 Answers
            2






            active

            oldest

            votes









            active

            oldest

            votes






            active

            oldest

            votes








            up vote
            5
            down vote



            accepted










            Unfortunately it is not even unique up to isomorphism! Take, for instance, $(mathbbZ/2mathbbZ)times (mathbbZ/4mathbbZ)$.






            share|cite|improve this answer




















            • Good example. Another: the dihedral group $D_4$ of order $8$ has at least two subgroups of order $4$: the Klein $4$-group and the subgroup consisting of the rotations. They are also non-iso.
              – Randall
              1 hour ago















            up vote
            5
            down vote



            accepted










            Unfortunately it is not even unique up to isomorphism! Take, for instance, $(mathbbZ/2mathbbZ)times (mathbbZ/4mathbbZ)$.






            share|cite|improve this answer




















            • Good example. Another: the dihedral group $D_4$ of order $8$ has at least two subgroups of order $4$: the Klein $4$-group and the subgroup consisting of the rotations. They are also non-iso.
              – Randall
              1 hour ago













            up vote
            5
            down vote



            accepted







            up vote
            5
            down vote



            accepted






            Unfortunately it is not even unique up to isomorphism! Take, for instance, $(mathbbZ/2mathbbZ)times (mathbbZ/4mathbbZ)$.






            share|cite|improve this answer












            Unfortunately it is not even unique up to isomorphism! Take, for instance, $(mathbbZ/2mathbbZ)times (mathbbZ/4mathbbZ)$.







            share|cite|improve this answer












            share|cite|improve this answer



            share|cite|improve this answer










            answered 1 hour ago









            ThomasGrubb

            9,95011335




            9,95011335











            • Good example. Another: the dihedral group $D_4$ of order $8$ has at least two subgroups of order $4$: the Klein $4$-group and the subgroup consisting of the rotations. They are also non-iso.
              – Randall
              1 hour ago

















            • Good example. Another: the dihedral group $D_4$ of order $8$ has at least two subgroups of order $4$: the Klein $4$-group and the subgroup consisting of the rotations. They are also non-iso.
              – Randall
              1 hour ago
















            Good example. Another: the dihedral group $D_4$ of order $8$ has at least two subgroups of order $4$: the Klein $4$-group and the subgroup consisting of the rotations. They are also non-iso.
            – Randall
            1 hour ago





            Good example. Another: the dihedral group $D_4$ of order $8$ has at least two subgroups of order $4$: the Klein $4$-group and the subgroup consisting of the rotations. They are also non-iso.
            – Randall
            1 hour ago











            up vote
            1
            down vote













            No way is it unique (generally). The group $G=mathbbZ_2 times mathbbZ_2$ has three subgroups of index $2$: $mathbbZ_2 times0$, the vice versa, and $(0,0), (1,1)$. You can use the same idea to cook up plenty of other examples.






            share|cite|improve this answer


























              up vote
              1
              down vote













              No way is it unique (generally). The group $G=mathbbZ_2 times mathbbZ_2$ has three subgroups of index $2$: $mathbbZ_2 times0$, the vice versa, and $(0,0), (1,1)$. You can use the same idea to cook up plenty of other examples.






              share|cite|improve this answer
























                up vote
                1
                down vote










                up vote
                1
                down vote









                No way is it unique (generally). The group $G=mathbbZ_2 times mathbbZ_2$ has three subgroups of index $2$: $mathbbZ_2 times0$, the vice versa, and $(0,0), (1,1)$. You can use the same idea to cook up plenty of other examples.






                share|cite|improve this answer














                No way is it unique (generally). The group $G=mathbbZ_2 times mathbbZ_2$ has three subgroups of index $2$: $mathbbZ_2 times0$, the vice versa, and $(0,0), (1,1)$. You can use the same idea to cook up plenty of other examples.







                share|cite|improve this answer














                share|cite|improve this answer



                share|cite|improve this answer








                edited 1 hour ago

























                answered 1 hour ago









                Randall

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