Example of subgroup of Direct Product which is not in usual form [duplicate]












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  • subgroup of direct product of two groups

    2 answers




I wanted to find group $Atimes B$ such that it has subgroup which is not of form $Ctimes D$ where $C < A$ and $D <B$.

Actually I think this is wrong I tried to get proof. But my friend told this is true . that it possible to have group with such prpoerty



ANy suggestion is appreciated






group-theory examples-counterexamples







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marked as duplicate by J.-E. Pin, amWhy group-theory
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 Nov 20 '18 at 12:12


This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.















  • You can only say that a subgroup of $A times B$ is a subdirect product of $C times D$ where $C < A$ and $D < B$.
    – J.-E. Pin
    Nov 20 '18 at 9:01










  • @J.-E.Pin WHat is a "subdirect product"?
    – DonAntonio
    Nov 20 '18 at 10:34












  • @donantonio Here is a link: subdirect product
    – J.-E. Pin
    Nov 20 '18 at 10:49


















0















This question already has an answer here:




  • subgroup of direct product of two groups

    2 answers




I wanted to find group $Atimes B$ such that it has subgroup which is not of form $Ctimes D$ where $C < A$ and $D <B$.

Actually I think this is wrong I tried to get proof. But my friend told this is true . that it possible to have group with such prpoerty



ANy suggestion is appreciated






group-theory examples-counterexamples







share|cite|improve this question















marked as duplicate by J.-E. Pin, amWhy group-theory
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 Nov 20 '18 at 12:12


This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.















  • You can only say that a subgroup of $A times B$ is a subdirect product of $C times D$ where $C < A$ and $D < B$.
    – J.-E. Pin
    Nov 20 '18 at 9:01










  • @J.-E.Pin WHat is a "subdirect product"?
    – DonAntonio
    Nov 20 '18 at 10:34












  • @donantonio Here is a link: subdirect product
    – J.-E. Pin
    Nov 20 '18 at 10:49
















0












0








0








This question already has an answer here:




  • subgroup of direct product of two groups

    2 answers




I wanted to find group $Atimes B$ such that it has subgroup which is not of form $Ctimes D$ where $C < A$ and $D <B$.

Actually I think this is wrong I tried to get proof. But my friend told this is true . that it possible to have group with such prpoerty



ANy suggestion is appreciated






group-theory examples-counterexamples







share|cite|improve this question
















This question already has an answer here:




  • subgroup of direct product of two groups

    2 answers




I wanted to find group $Atimes B$ such that it has subgroup which is not of form $Ctimes D$ where $C < A$ and $D <B$.

Actually I think this is wrong I tried to get proof. But my friend told this is true . that it possible to have group with such prpoerty



ANy suggestion is appreciated





This question already has an answer here:




  • subgroup of direct product of two groups

    2 answers







group-theory examples-counterexamples




group-theory examples-counterexamples






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




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edited Nov 20 '18 at 8:29

























asked Nov 20 '18 at 8:24









Shubham

1,5851519




1,5851519




marked as duplicate by J.-E. Pin, amWhy group-theory
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 Nov 20 '18 at 12:12


This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.














  • You can only say that a subgroup of $A times B$ is a subdirect product of $C times D$ where $C < A$ and $D < B$.
    – J.-E. Pin
    Nov 20 '18 at 9:01










  • @J.-E.Pin WHat is a "subdirect product"?
    – DonAntonio
    Nov 20 '18 at 10:34












  • @donantonio Here is a link: subdirect product
    – J.-E. Pin
    Nov 20 '18 at 10:49




















  • You can only say that a subgroup of $A times B$ is a subdirect product of $C times D$ where $C < A$ and $D < B$.
    – J.-E. Pin
    Nov 20 '18 at 9:01










  • @J.-E.Pin WHat is a "subdirect product"?
    – DonAntonio
    Nov 20 '18 at 10:34












  • @donantonio Here is a link: subdirect product
    – J.-E. Pin
    Nov 20 '18 at 10:49


















You can only say that a subgroup of $A times B$ is a subdirect product of $C times D$ where $C < A$ and $D < B$.
– J.-E. Pin
Nov 20 '18 at 9:01




You can only say that a subgroup of $A times B$ is a subdirect product of $C times D$ where $C < A$ and $D < B$.
– J.-E. Pin
Nov 20 '18 at 9:01












@J.-E.Pin WHat is a "subdirect product"?
– DonAntonio
Nov 20 '18 at 10:34






@J.-E.Pin WHat is a "subdirect product"?
– DonAntonio
Nov 20 '18 at 10:34














@donantonio Here is a link: subdirect product
– J.-E. Pin
Nov 20 '18 at 10:49






@donantonio Here is a link: subdirect product
– J.-E. Pin
Nov 20 '18 at 10:49












1 Answer
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Consider $G=Bbb{Z}_2 times Bbb{Z}_2$ and let $H=langle (1,1) rangle={(0,0), (1,1)}$. Then $H leq G$ but $H$ cannot be expressed as $C times D$. Because if it could be, then $0,1 in C$ and $0,1 in D$. This will mean $C=Bbb{Z}_2$ and same for $D$. But then $C times D$ will have $4$ elements. Thus cannot be same as $H$.






share|cite|improve this answer




























    1 Answer
    1






    active

    oldest

    votes








    1 Answer
    1






    active

    oldest

    votes









    active

    oldest

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    active

    oldest

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    3














    Consider $G=Bbb{Z}_2 times Bbb{Z}_2$ and let $H=langle (1,1) rangle={(0,0), (1,1)}$. Then $H leq G$ but $H$ cannot be expressed as $C times D$. Because if it could be, then $0,1 in C$ and $0,1 in D$. This will mean $C=Bbb{Z}_2$ and same for $D$. But then $C times D$ will have $4$ elements. Thus cannot be same as $H$.






    share|cite|improve this answer


























      3














      Consider $G=Bbb{Z}_2 times Bbb{Z}_2$ and let $H=langle (1,1) rangle={(0,0), (1,1)}$. Then $H leq G$ but $H$ cannot be expressed as $C times D$. Because if it could be, then $0,1 in C$ and $0,1 in D$. This will mean $C=Bbb{Z}_2$ and same for $D$. But then $C times D$ will have $4$ elements. Thus cannot be same as $H$.






      share|cite|improve this answer
























        3












        3








        3






        Consider $G=Bbb{Z}_2 times Bbb{Z}_2$ and let $H=langle (1,1) rangle={(0,0), (1,1)}$. Then $H leq G$ but $H$ cannot be expressed as $C times D$. Because if it could be, then $0,1 in C$ and $0,1 in D$. This will mean $C=Bbb{Z}_2$ and same for $D$. But then $C times D$ will have $4$ elements. Thus cannot be same as $H$.






        share|cite|improve this answer












        Consider $G=Bbb{Z}_2 times Bbb{Z}_2$ and let $H=langle (1,1) rangle={(0,0), (1,1)}$. Then $H leq G$ but $H$ cannot be expressed as $C times D$. Because if it could be, then $0,1 in C$ and $0,1 in D$. This will mean $C=Bbb{Z}_2$ and same for $D$. But then $C times D$ will have $4$ elements. Thus cannot be same as $H$.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Nov 20 '18 at 8:29









        Anurag A

        25.6k12249




        25.6k12249















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