Cyclic normal subgroups











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I have a question: it is true that if $H_1$ and $H_2$ are cyclic groups that are isomorphic, then $G/H_1$ is isomorphic to $G/H_2$? I know that if I remove the condition "cyclic groups", the given statement is false and there are numerous counterexamples that disproves it, but I don't know if my statement is true and I don't how to create a counterexample or to prove it. If it is true, can you give me a hint about how can this be proven?



For example, I just have shown that $Z_{12}/ langle 2 rangle$ is isomorphic to $Z_{12}/Z_6$ which is isomorphic to $Z_3$ (since both $langle 2 rangle$ and $Z_6$ are isomorphic). How this can be generalized?






group-theory normal-subgroups







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    up vote
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    favorite












    I have a question: it is true that if $H_1$ and $H_2$ are cyclic groups that are isomorphic, then $G/H_1$ is isomorphic to $G/H_2$? I know that if I remove the condition "cyclic groups", the given statement is false and there are numerous counterexamples that disproves it, but I don't know if my statement is true and I don't how to create a counterexample or to prove it. If it is true, can you give me a hint about how can this be proven?



    For example, I just have shown that $Z_{12}/ langle 2 rangle$ is isomorphic to $Z_{12}/Z_6$ which is isomorphic to $Z_3$ (since both $langle 2 rangle$ and $Z_6$ are isomorphic). How this can be generalized?






    group-theory normal-subgroups







    share|cite|improve this question


























      up vote
      2
      down vote

      favorite









      up vote
      2
      down vote

      favorite











      I have a question: it is true that if $H_1$ and $H_2$ are cyclic groups that are isomorphic, then $G/H_1$ is isomorphic to $G/H_2$? I know that if I remove the condition "cyclic groups", the given statement is false and there are numerous counterexamples that disproves it, but I don't know if my statement is true and I don't how to create a counterexample or to prove it. If it is true, can you give me a hint about how can this be proven?



      For example, I just have shown that $Z_{12}/ langle 2 rangle$ is isomorphic to $Z_{12}/Z_6$ which is isomorphic to $Z_3$ (since both $langle 2 rangle$ and $Z_6$ are isomorphic). How this can be generalized?






      group-theory normal-subgroups







      share|cite|improve this question















      I have a question: it is true that if $H_1$ and $H_2$ are cyclic groups that are isomorphic, then $G/H_1$ is isomorphic to $G/H_2$? I know that if I remove the condition "cyclic groups", the given statement is false and there are numerous counterexamples that disproves it, but I don't know if my statement is true and I don't how to create a counterexample or to prove it. If it is true, can you give me a hint about how can this be proven?



      For example, I just have shown that $Z_{12}/ langle 2 rangle$ is isomorphic to $Z_{12}/Z_6$ which is isomorphic to $Z_3$ (since both $langle 2 rangle$ and $Z_6$ are isomorphic). How this can be generalized?






      group-theory normal-subgroups




      group-theory normal-subgroups






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      share|cite|improve this question













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      edited 4 hours ago









      the_fox

      2,0711429




      2,0711429










      asked 6 hours ago









      user573497

      16419




      16419






















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          No, that's not true. Take $G = C_2 times C_4$ and note that since $G$ is abelian, every subgroup of $G$ is normal in $G$. Let $H = C_2$ be the first direct factor of $G$ and $K$ be the unique subgroup of order $2$ of $C_4$. Obviously, $H cong K$ since both have order $2$, but $G/H cong C_4$ while $G/K cong C_2 times C_2$.






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            Did you try cyclic subgroups of the additive group of integers $mathbb{Z}$?






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            • But what about integers mod n? I edited my question and I found very interesting that observation about the isomorphism about quotient groups.
              – user573497
              6 hours ago











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            No, that's not true. Take $G = C_2 times C_4$ and note that since $G$ is abelian, every subgroup of $G$ is normal in $G$. Let $H = C_2$ be the first direct factor of $G$ and $K$ be the unique subgroup of order $2$ of $C_4$. Obviously, $H cong K$ since both have order $2$, but $G/H cong C_4$ while $G/K cong C_2 times C_2$.






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              No, that's not true. Take $G = C_2 times C_4$ and note that since $G$ is abelian, every subgroup of $G$ is normal in $G$. Let $H = C_2$ be the first direct factor of $G$ and $K$ be the unique subgroup of order $2$ of $C_4$. Obviously, $H cong K$ since both have order $2$, but $G/H cong C_4$ while $G/K cong C_2 times C_2$.






              share|cite|improve this answer























                up vote
                4
                down vote










                up vote
                4
                down vote









                No, that's not true. Take $G = C_2 times C_4$ and note that since $G$ is abelian, every subgroup of $G$ is normal in $G$. Let $H = C_2$ be the first direct factor of $G$ and $K$ be the unique subgroup of order $2$ of $C_4$. Obviously, $H cong K$ since both have order $2$, but $G/H cong C_4$ while $G/K cong C_2 times C_2$.






                share|cite|improve this answer












                No, that's not true. Take $G = C_2 times C_4$ and note that since $G$ is abelian, every subgroup of $G$ is normal in $G$. Let $H = C_2$ be the first direct factor of $G$ and $K$ be the unique subgroup of order $2$ of $C_4$. Obviously, $H cong K$ since both have order $2$, but $G/H cong C_4$ while $G/K cong C_2 times C_2$.







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered 6 hours ago









                the_fox

                2,0711429




                2,0711429






















                    up vote
                    2
                    down vote













                    Did you try cyclic subgroups of the additive group of integers $mathbb{Z}$?






                    share|cite|improve this answer





















                    • But what about integers mod n? I edited my question and I found very interesting that observation about the isomorphism about quotient groups.
                      – user573497
                      6 hours ago















                    up vote
                    2
                    down vote













                    Did you try cyclic subgroups of the additive group of integers $mathbb{Z}$?






                    share|cite|improve this answer





















                    • But what about integers mod n? I edited my question and I found very interesting that observation about the isomorphism about quotient groups.
                      – user573497
                      6 hours ago













                    up vote
                    2
                    down vote










                    up vote
                    2
                    down vote









                    Did you try cyclic subgroups of the additive group of integers $mathbb{Z}$?






                    share|cite|improve this answer












                    Did you try cyclic subgroups of the additive group of integers $mathbb{Z}$?







                    share|cite|improve this answer












                    share|cite|improve this answer



                    share|cite|improve this answer










                    answered 6 hours ago









                    Bartosz Malman

                    6881520




                    6881520












                    • But what about integers mod n? I edited my question and I found very interesting that observation about the isomorphism about quotient groups.
                      – user573497
                      6 hours ago


















                    • But what about integers mod n? I edited my question and I found very interesting that observation about the isomorphism about quotient groups.
                      – user573497
                      6 hours ago
















                    But what about integers mod n? I edited my question and I found very interesting that observation about the isomorphism about quotient groups.
                    – user573497
                    6 hours ago




                    But what about integers mod n? I edited my question and I found very interesting that observation about the isomorphism about quotient groups.
                    – user573497
                    6 hours ago


















                     

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