Number of generators of subgroup Announcing the arrival of Valued Associate #679: Cesar Manara Planned maintenance scheduled April 23, 2019 at 23:30 UTC (7:30pm US/Eastern)Torsion subgroupOn the minimal number of generators of a finite groupBound number of generators of a subgroup of a nilpotent group?Minimal number of generators for a finitely generated abelian $p$-groupA question on finitely generated Abelian groups with a minimal number of generatorsFactoring an Abelian groupThe number of internal direct summands of a finitely generated abelian groupFree group generated by two generators is isomorphic to product of two infinite cyclic groupsAlternative proof of the Fundamental Theorem of Abelian Groups??Hungerford Chapter 2 Section 2 Problem 2 WITHOUT using the structure theorem of finite abelian groups

How does TikZ render an arc?

Where and when has Thucydides been studied?

French equivalents of おしゃれは足元から (Every good outfit starts with the shoes)

Where did Ptolemy compare the Earth to the distance of fixed stars?

Did pre-Columbian Americans know the spherical shape of the Earth?

Pointing to problems without suggesting solutions

Is this Half-dragon Quaggoth boss monster balanced?

What does 丫 mean? 丫是什么意思?

Can two people see the same photon?

Is it OK to use the testing sample to compare algorithms?

3D Masyu - A Die

An isoperimetric-type inequality inside a cube

Are there any irrational/transcendental numbers for which the distribution of decimal digits is not uniform?

Weaponising the Grasp-at-a-Distance spell

Keep at all times, the minus sign above aligned with minus sign below

malloc in main() or malloc in another function: allocating memory for a struct and its members

The test team as an enemy of development? And how can this be avoided?

Short story about astronauts fertilizing soil with their own bodies

Order between one to one functions and their inverses

Is there any significance to the prison numbers of the Beagle Boys starting with 176-?

Did John Wesley plagiarize Matthew Henry...?

Table formatting with tabularx?

How much damage would a cupful of neutron star matter do to the Earth?

Problem with display of presentation



Number of generators of subgroup



Announcing the arrival of Valued Associate #679: Cesar Manara
Planned maintenance scheduled April 23, 2019 at 23:30 UTC (7:30pm US/Eastern)Torsion subgroupOn the minimal number of generators of a finite groupBound number of generators of a subgroup of a nilpotent group?Minimal number of generators for a finitely generated abelian $p$-groupA question on finitely generated Abelian groups with a minimal number of generatorsFactoring an Abelian groupThe number of internal direct summands of a finitely generated abelian groupFree group generated by two generators is isomorphic to product of two infinite cyclic groupsAlternative proof of the Fundamental Theorem of Abelian Groups??Hungerford Chapter 2 Section 2 Problem 2 WITHOUT using the structure theorem of finite abelian groups










1












$begingroup$


I am trying to prove the following.



let $G$ be a finitely generated abelian group, and $H<G$ a subgroup such that there exists a subgroup $K<G$ and we can write $G=H oplus K$. Is it true that the minimal number of generators of H is strictly smaller than the minimal number of generators of $G$?



Clearly if G can not be written as a direct summand of $H$ then this is not true, just consider $G= mathbbZ$ and $H=2mathbbZ$.



I would like to prove it because I believe it can provide a simpler proof for the characterization of finitely generated abelian groups.










share|cite|improve this question











$endgroup$











  • $begingroup$
    $mathbb Zbig / 2mathbb Z oplus mathbb Zbig / 3mathbb Z $ is cyclic.
    $endgroup$
    – lulu
    5 hours ago






  • 2




    $begingroup$
    Worth noting: "number of generators" is not well defined. I'm guessing you mean "minimal number of generators", but you should say so,
    $endgroup$
    – lulu
    5 hours ago










  • $begingroup$
    Thank you for pointing that out. I will edit to correct it.
    $endgroup$
    – Charles
    5 hours ago















1












$begingroup$


I am trying to prove the following.



let $G$ be a finitely generated abelian group, and $H<G$ a subgroup such that there exists a subgroup $K<G$ and we can write $G=H oplus K$. Is it true that the minimal number of generators of H is strictly smaller than the minimal number of generators of $G$?



Clearly if G can not be written as a direct summand of $H$ then this is not true, just consider $G= mathbbZ$ and $H=2mathbbZ$.



I would like to prove it because I believe it can provide a simpler proof for the characterization of finitely generated abelian groups.










share|cite|improve this question











$endgroup$











  • $begingroup$
    $mathbb Zbig / 2mathbb Z oplus mathbb Zbig / 3mathbb Z $ is cyclic.
    $endgroup$
    – lulu
    5 hours ago






  • 2




    $begingroup$
    Worth noting: "number of generators" is not well defined. I'm guessing you mean "minimal number of generators", but you should say so,
    $endgroup$
    – lulu
    5 hours ago










  • $begingroup$
    Thank you for pointing that out. I will edit to correct it.
    $endgroup$
    – Charles
    5 hours ago













1












1








1





$begingroup$


I am trying to prove the following.



let $G$ be a finitely generated abelian group, and $H<G$ a subgroup such that there exists a subgroup $K<G$ and we can write $G=H oplus K$. Is it true that the minimal number of generators of H is strictly smaller than the minimal number of generators of $G$?



Clearly if G can not be written as a direct summand of $H$ then this is not true, just consider $G= mathbbZ$ and $H=2mathbbZ$.



I would like to prove it because I believe it can provide a simpler proof for the characterization of finitely generated abelian groups.










share|cite|improve this question











$endgroup$




I am trying to prove the following.



let $G$ be a finitely generated abelian group, and $H<G$ a subgroup such that there exists a subgroup $K<G$ and we can write $G=H oplus K$. Is it true that the minimal number of generators of H is strictly smaller than the minimal number of generators of $G$?



Clearly if G can not be written as a direct summand of $H$ then this is not true, just consider $G= mathbbZ$ and $H=2mathbbZ$.



I would like to prove it because I believe it can provide a simpler proof for the characterization of finitely generated abelian groups.







group-theory abelian-groups






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited 4 hours ago







Charles

















asked 5 hours ago









CharlesCharles

582420




582420











  • $begingroup$
    $mathbb Zbig / 2mathbb Z oplus mathbb Zbig / 3mathbb Z $ is cyclic.
    $endgroup$
    – lulu
    5 hours ago






  • 2




    $begingroup$
    Worth noting: "number of generators" is not well defined. I'm guessing you mean "minimal number of generators", but you should say so,
    $endgroup$
    – lulu
    5 hours ago










  • $begingroup$
    Thank you for pointing that out. I will edit to correct it.
    $endgroup$
    – Charles
    5 hours ago
















  • $begingroup$
    $mathbb Zbig / 2mathbb Z oplus mathbb Zbig / 3mathbb Z $ is cyclic.
    $endgroup$
    – lulu
    5 hours ago






  • 2




    $begingroup$
    Worth noting: "number of generators" is not well defined. I'm guessing you mean "minimal number of generators", but you should say so,
    $endgroup$
    – lulu
    5 hours ago










  • $begingroup$
    Thank you for pointing that out. I will edit to correct it.
    $endgroup$
    – Charles
    5 hours ago















$begingroup$
$mathbb Zbig / 2mathbb Z oplus mathbb Zbig / 3mathbb Z $ is cyclic.
$endgroup$
– lulu
5 hours ago




$begingroup$
$mathbb Zbig / 2mathbb Z oplus mathbb Zbig / 3mathbb Z $ is cyclic.
$endgroup$
– lulu
5 hours ago




2




2




$begingroup$
Worth noting: "number of generators" is not well defined. I'm guessing you mean "minimal number of generators", but you should say so,
$endgroup$
– lulu
5 hours ago




$begingroup$
Worth noting: "number of generators" is not well defined. I'm guessing you mean "minimal number of generators", but you should say so,
$endgroup$
– lulu
5 hours ago












$begingroup$
Thank you for pointing that out. I will edit to correct it.
$endgroup$
– Charles
5 hours ago




$begingroup$
Thank you for pointing that out. I will edit to correct it.
$endgroup$
– Charles
5 hours ago










1 Answer
1






active

oldest

votes


















4












$begingroup$

No, it is not true. Consider $mathbbZ_2oplusmathbbZ_3$. This has a generator $(1,1)$. Note that
$$0oplusmathbbZ_3<mathbbZ_2oplusmathbbZ_3 ,$$
and
$$(mathbbZ_2oplus 0)oplus(0oplusmathbbZ_3)=mathbbZ_2oplusmathbbZ_3.$$
However, $0oplusmathbbZ_3$ is generated by $(0,1).$






share|cite|improve this answer









$endgroup$













    Your Answer








    StackExchange.ready(function()
    var channelOptions =
    tags: "".split(" "),
    id: "69"
    ;
    initTagRenderer("".split(" "), "".split(" "), channelOptions);

    StackExchange.using("externalEditor", function()
    // Have to fire editor after snippets, if snippets enabled
    if (StackExchange.settings.snippets.snippetsEnabled)
    StackExchange.using("snippets", function()
    createEditor();
    );

    else
    createEditor();

    );

    function createEditor()
    StackExchange.prepareEditor(
    heartbeatType: 'answer',
    autoActivateHeartbeat: false,
    convertImagesToLinks: true,
    noModals: true,
    showLowRepImageUploadWarning: true,
    reputationToPostImages: 10,
    bindNavPrevention: true,
    postfix: "",
    imageUploader:
    brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
    contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
    allowUrls: true
    ,
    noCode: true, onDemand: true,
    discardSelector: ".discard-answer"
    ,immediatelyShowMarkdownHelp:true
    );



    );













    draft saved

    draft discarded


















    StackExchange.ready(
    function ()
    StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3196206%2fnumber-of-generators-of-subgroup%23new-answer', 'question_page');

    );

    Post as a guest















    Required, but never shown

























    1 Answer
    1






    active

    oldest

    votes








    1 Answer
    1






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    4












    $begingroup$

    No, it is not true. Consider $mathbbZ_2oplusmathbbZ_3$. This has a generator $(1,1)$. Note that
    $$0oplusmathbbZ_3<mathbbZ_2oplusmathbbZ_3 ,$$
    and
    $$(mathbbZ_2oplus 0)oplus(0oplusmathbbZ_3)=mathbbZ_2oplusmathbbZ_3.$$
    However, $0oplusmathbbZ_3$ is generated by $(0,1).$






    share|cite|improve this answer









    $endgroup$

















      4












      $begingroup$

      No, it is not true. Consider $mathbbZ_2oplusmathbbZ_3$. This has a generator $(1,1)$. Note that
      $$0oplusmathbbZ_3<mathbbZ_2oplusmathbbZ_3 ,$$
      and
      $$(mathbbZ_2oplus 0)oplus(0oplusmathbbZ_3)=mathbbZ_2oplusmathbbZ_3.$$
      However, $0oplusmathbbZ_3$ is generated by $(0,1).$






      share|cite|improve this answer









      $endgroup$















        4












        4








        4





        $begingroup$

        No, it is not true. Consider $mathbbZ_2oplusmathbbZ_3$. This has a generator $(1,1)$. Note that
        $$0oplusmathbbZ_3<mathbbZ_2oplusmathbbZ_3 ,$$
        and
        $$(mathbbZ_2oplus 0)oplus(0oplusmathbbZ_3)=mathbbZ_2oplusmathbbZ_3.$$
        However, $0oplusmathbbZ_3$ is generated by $(0,1).$






        share|cite|improve this answer









        $endgroup$



        No, it is not true. Consider $mathbbZ_2oplusmathbbZ_3$. This has a generator $(1,1)$. Note that
        $$0oplusmathbbZ_3<mathbbZ_2oplusmathbbZ_3 ,$$
        and
        $$(mathbbZ_2oplus 0)oplus(0oplusmathbbZ_3)=mathbbZ_2oplusmathbbZ_3.$$
        However, $0oplusmathbbZ_3$ is generated by $(0,1).$







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered 5 hours ago









        MelodyMelody

        1,41212




        1,41212



























            draft saved

            draft discarded
















































            Thanks for contributing an answer to Mathematics Stack Exchange!


            • Please be sure to answer the question. Provide details and share your research!

            But avoid


            • Asking for help, clarification, or responding to other answers.

            • Making statements based on opinion; back them up with references or personal experience.

            Use MathJax to format equations. MathJax reference.


            To learn more, see our tips on writing great answers.




            draft saved


            draft discarded














            StackExchange.ready(
            function ()
            StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3196206%2fnumber-of-generators-of-subgroup%23new-answer', 'question_page');

            );

            Post as a guest















            Required, but never shown





















































            Required, but never shown














            Required, but never shown












            Required, but never shown







            Required, but never shown

































            Required, but never shown














            Required, but never shown












            Required, but never shown







            Required, but never shown







            Popular posts from this blog

            名間水力發電廠 目录 沿革 設施 鄰近設施 註釋 外部連結 导航菜单23°50′10″N 120°42′41″E / 23.83611°N 120.71139°E / 23.83611; 120.7113923°50′10″N 120°42′41″E / 23.83611°N 120.71139°E / 23.83611; 120.71139計畫概要原始内容臺灣第一座BOT 模式開發的水力發電廠-名間水力電廠名間水力發電廠 水利署首件BOT案原始内容《小檔案》名間電廠 首座BOT水力發電廠原始内容名間電廠BOT - 經濟部水利署中區水資源局

            Prove that NP is closed under karp reduction?Space(n) not closed under Karp reductions - what about NTime(n)?Class P is closed under rotation?Prove or disprove that $NL$ is closed under polynomial many-one reductions$mathbfNC_2$ is closed under log-space reductionOn Karp reductionwhen can I know if a class (complexity) is closed under reduction (cook/karp)Check if class $PSPACE$ is closed under polyonomially space reductionIs NPSPACE also closed under polynomial-time reduction and under log-space reduction?Prove PSPACE is closed under complement?Prove PSPACE is closed under union?

            Is my guitar’s action too high? Announcing the arrival of Valued Associate #679: Cesar Manara Planned maintenance scheduled April 23, 2019 at 23:30 UTC (7:30pm US/Eastern)Strings too stiff on a recently purchased acoustic guitar | Cort AD880CEIs the action of my guitar really high?Μy little finger is too weak to play guitarWith guitar, how long should I give my fingers to strengthen / callous?When playing a fret the guitar sounds mutedPlaying (Barre) chords up the guitar neckI think my guitar strings are wound too tight and I can't play barre chordsF barre chord on an SG guitarHow to find to the right strings of a barre chord by feel?High action on higher fret on my steel acoustic guitar