Presentation of wreath product $G=S_3 wr S_3$ of symmetric groups. What is the isomorphism type of $G/[G,G]$?












4












$begingroup$


I'm trying to answer the first part of a group theory question as revision for an exam that goes as follows;




Let $G = S_3 wr S_3$, the permutational wreath product of two symmetric groups of degree three. Give a presentation for $G$ and determine the isomorphism type of $G/[G, G]$.




I'm not sure how to go about finding generators for the wreath product itself.




Is there a method for combining the generators of the symmetric groups to form generators for the wreath prouct?




Any pointers would be much appreciated, thanks in advance!






group-theory finite-groups group-presentation combinatorial-group-theory wreath-product







share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    If you can't find generators, then you haven't got much hope of answering the question! The obvious generating set consists of generators of the two natural subgroups isomorphic to $S_3$, giving four generators in all. For these,you could take for example: $(1,2,3)$, $(2,3)$, $(1,4,7)(2,5,8)(3,6,9)$, and $(4,7)(5,8)(6,9)$.
    $endgroup$
    – Derek Holt
    Apr 20 '17 at 18:29












  • $begingroup$
    I would like to know what a presentation of the Wreath product of two groups given by $langle Xmid Rrangle$ and $langle Ymid Srangle$ looks like in general, if possible; I might ask a question on MSE about it.
    $endgroup$
    – Shaun
    Nov 29 '18 at 23:23










  • $begingroup$
    @DerekHolt, do you know of such a presentation? I'm aware of, say, this, but, for some reason, I can't get access to it; my institution is not subscribed :(
    $endgroup$
    – Shaun
    Nov 29 '18 at 23:26










  • $begingroup$
    @shaun I have answered the question.
    $endgroup$
    – Derek Holt
    Nov 30 '18 at 8:12










  • $begingroup$
    Thank you, @DerekHolt.
    $endgroup$
    – Shaun
    Dec 1 '18 at 3:37
















4












$begingroup$


I'm trying to answer the first part of a group theory question as revision for an exam that goes as follows;




Let $G = S_3 wr S_3$, the permutational wreath product of two symmetric groups of degree three. Give a presentation for $G$ and determine the isomorphism type of $G/[G, G]$.




I'm not sure how to go about finding generators for the wreath product itself.




Is there a method for combining the generators of the symmetric groups to form generators for the wreath prouct?




Any pointers would be much appreciated, thanks in advance!






group-theory finite-groups group-presentation combinatorial-group-theory wreath-product







share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    If you can't find generators, then you haven't got much hope of answering the question! The obvious generating set consists of generators of the two natural subgroups isomorphic to $S_3$, giving four generators in all. For these,you could take for example: $(1,2,3)$, $(2,3)$, $(1,4,7)(2,5,8)(3,6,9)$, and $(4,7)(5,8)(6,9)$.
    $endgroup$
    – Derek Holt
    Apr 20 '17 at 18:29












  • $begingroup$
    I would like to know what a presentation of the Wreath product of two groups given by $langle Xmid Rrangle$ and $langle Ymid Srangle$ looks like in general, if possible; I might ask a question on MSE about it.
    $endgroup$
    – Shaun
    Nov 29 '18 at 23:23










  • $begingroup$
    @DerekHolt, do you know of such a presentation? I'm aware of, say, this, but, for some reason, I can't get access to it; my institution is not subscribed :(
    $endgroup$
    – Shaun
    Nov 29 '18 at 23:26










  • $begingroup$
    @shaun I have answered the question.
    $endgroup$
    – Derek Holt
    Nov 30 '18 at 8:12










  • $begingroup$
    Thank you, @DerekHolt.
    $endgroup$
    – Shaun
    Dec 1 '18 at 3:37














4












4








4


2



$begingroup$


I'm trying to answer the first part of a group theory question as revision for an exam that goes as follows;




Let $G = S_3 wr S_3$, the permutational wreath product of two symmetric groups of degree three. Give a presentation for $G$ and determine the isomorphism type of $G/[G, G]$.




I'm not sure how to go about finding generators for the wreath product itself.




Is there a method for combining the generators of the symmetric groups to form generators for the wreath prouct?




Any pointers would be much appreciated, thanks in advance!






group-theory finite-groups group-presentation combinatorial-group-theory wreath-product







share|cite|improve this question











$endgroup$




I'm trying to answer the first part of a group theory question as revision for an exam that goes as follows;




Let $G = S_3 wr S_3$, the permutational wreath product of two symmetric groups of degree three. Give a presentation for $G$ and determine the isomorphism type of $G/[G, G]$.




I'm not sure how to go about finding generators for the wreath product itself.




Is there a method for combining the generators of the symmetric groups to form generators for the wreath prouct?




Any pointers would be much appreciated, thanks in advance!






group-theory finite-groups group-presentation combinatorial-group-theory wreath-product




group-theory finite-groups group-presentation combinatorial-group-theory wreath-product






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Nov 29 '18 at 23:05









Shaun

9,083113683




9,083113683










asked Apr 20 '17 at 14:45









AidanpmAidanpm

212




212








  • 1




    $begingroup$
    If you can't find generators, then you haven't got much hope of answering the question! The obvious generating set consists of generators of the two natural subgroups isomorphic to $S_3$, giving four generators in all. For these,you could take for example: $(1,2,3)$, $(2,3)$, $(1,4,7)(2,5,8)(3,6,9)$, and $(4,7)(5,8)(6,9)$.
    $endgroup$
    – Derek Holt
    Apr 20 '17 at 18:29












  • $begingroup$
    I would like to know what a presentation of the Wreath product of two groups given by $langle Xmid Rrangle$ and $langle Ymid Srangle$ looks like in general, if possible; I might ask a question on MSE about it.
    $endgroup$
    – Shaun
    Nov 29 '18 at 23:23










  • $begingroup$
    @DerekHolt, do you know of such a presentation? I'm aware of, say, this, but, for some reason, I can't get access to it; my institution is not subscribed :(
    $endgroup$
    – Shaun
    Nov 29 '18 at 23:26










  • $begingroup$
    @shaun I have answered the question.
    $endgroup$
    – Derek Holt
    Nov 30 '18 at 8:12










  • $begingroup$
    Thank you, @DerekHolt.
    $endgroup$
    – Shaun
    Dec 1 '18 at 3:37














  • 1




    $begingroup$
    If you can't find generators, then you haven't got much hope of answering the question! The obvious generating set consists of generators of the two natural subgroups isomorphic to $S_3$, giving four generators in all. For these,you could take for example: $(1,2,3)$, $(2,3)$, $(1,4,7)(2,5,8)(3,6,9)$, and $(4,7)(5,8)(6,9)$.
    $endgroup$
    – Derek Holt
    Apr 20 '17 at 18:29












  • $begingroup$
    I would like to know what a presentation of the Wreath product of two groups given by $langle Xmid Rrangle$ and $langle Ymid Srangle$ looks like in general, if possible; I might ask a question on MSE about it.
    $endgroup$
    – Shaun
    Nov 29 '18 at 23:23










  • $begingroup$
    @DerekHolt, do you know of such a presentation? I'm aware of, say, this, but, for some reason, I can't get access to it; my institution is not subscribed :(
    $endgroup$
    – Shaun
    Nov 29 '18 at 23:26










  • $begingroup$
    @shaun I have answered the question.
    $endgroup$
    – Derek Holt
    Nov 30 '18 at 8:12










  • $begingroup$
    Thank you, @DerekHolt.
    $endgroup$
    – Shaun
    Dec 1 '18 at 3:37








1




1




$begingroup$
If you can't find generators, then you haven't got much hope of answering the question! The obvious generating set consists of generators of the two natural subgroups isomorphic to $S_3$, giving four generators in all. For these,you could take for example: $(1,2,3)$, $(2,3)$, $(1,4,7)(2,5,8)(3,6,9)$, and $(4,7)(5,8)(6,9)$.
$endgroup$
– Derek Holt
Apr 20 '17 at 18:29






$begingroup$
If you can't find generators, then you haven't got much hope of answering the question! The obvious generating set consists of generators of the two natural subgroups isomorphic to $S_3$, giving four generators in all. For these,you could take for example: $(1,2,3)$, $(2,3)$, $(1,4,7)(2,5,8)(3,6,9)$, and $(4,7)(5,8)(6,9)$.
$endgroup$
– Derek Holt
Apr 20 '17 at 18:29














$begingroup$
I would like to know what a presentation of the Wreath product of two groups given by $langle Xmid Rrangle$ and $langle Ymid Srangle$ looks like in general, if possible; I might ask a question on MSE about it.
$endgroup$
– Shaun
Nov 29 '18 at 23:23




$begingroup$
I would like to know what a presentation of the Wreath product of two groups given by $langle Xmid Rrangle$ and $langle Ymid Srangle$ looks like in general, if possible; I might ask a question on MSE about it.
$endgroup$
– Shaun
Nov 29 '18 at 23:23












$begingroup$
@DerekHolt, do you know of such a presentation? I'm aware of, say, this, but, for some reason, I can't get access to it; my institution is not subscribed :(
$endgroup$
– Shaun
Nov 29 '18 at 23:26




$begingroup$
@DerekHolt, do you know of such a presentation? I'm aware of, say, this, but, for some reason, I can't get access to it; my institution is not subscribed :(
$endgroup$
– Shaun
Nov 29 '18 at 23:26












$begingroup$
@shaun I have answered the question.
$endgroup$
– Derek Holt
Nov 30 '18 at 8:12




$begingroup$
@shaun I have answered the question.
$endgroup$
– Derek Holt
Nov 30 '18 at 8:12












$begingroup$
Thank you, @DerekHolt.
$endgroup$
– Shaun
Dec 1 '18 at 3:37




$begingroup$
Thank you, @DerekHolt.
$endgroup$
– Shaun
Dec 1 '18 at 3:37










1 Answer
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$begingroup$

Here is a presentation of $S_3 wr S_3$. Note that $a,b$ generate the top $S_3$ factor, and $x,y$ generate one of the three factors of the base group, the other two being $langle x^a,y^a rangle$ and $langle x^{a^2},y^{a^2}rangle$, where $x^a = a^{-1}xa$.
$$langle a,b,x,y,mid a^3,b^2,(ab)^2,x^3,y^2,(xy)^2,
[x,x^a],[y,x^a],[x,x^{a^2}],[y,x^{a^2}],
[x,y^a],[y,y^a],[x,y^{a^2}],[y,y^{a^2}],
[x,b], [y,b] rangle.$$
A routine calculation from the presentation shows that $G/[G,G] cong C_2 times C_2$, but it is not hard to show that directly.



PS: In fact the four relators $[x,x^{a^2}],[y,x^{a^2}],[x,y^{a^2}],[y,y^{a^2}]$ are redundant. The fact that $langle x,y rangle$ commutes with $langle x^{a^2},y^{a^2} rangle$ follows from the fact that $langle x,y rangle$ commutes with $langle x^{a},y^{a} rangle$ by conjugating by $b$.






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    $begingroup$

    Here is a presentation of $S_3 wr S_3$. Note that $a,b$ generate the top $S_3$ factor, and $x,y$ generate one of the three factors of the base group, the other two being $langle x^a,y^a rangle$ and $langle x^{a^2},y^{a^2}rangle$, where $x^a = a^{-1}xa$.
    $$langle a,b,x,y,mid a^3,b^2,(ab)^2,x^3,y^2,(xy)^2,
    [x,x^a],[y,x^a],[x,x^{a^2}],[y,x^{a^2}],
    [x,y^a],[y,y^a],[x,y^{a^2}],[y,y^{a^2}],
    [x,b], [y,b] rangle.$$
    A routine calculation from the presentation shows that $G/[G,G] cong C_2 times C_2$, but it is not hard to show that directly.



    PS: In fact the four relators $[x,x^{a^2}],[y,x^{a^2}],[x,y^{a^2}],[y,y^{a^2}]$ are redundant. The fact that $langle x,y rangle$ commutes with $langle x^{a^2},y^{a^2} rangle$ follows from the fact that $langle x,y rangle$ commutes with $langle x^{a},y^{a} rangle$ by conjugating by $b$.






    share|cite|improve this answer











    $endgroup$


















      3












      $begingroup$

      Here is a presentation of $S_3 wr S_3$. Note that $a,b$ generate the top $S_3$ factor, and $x,y$ generate one of the three factors of the base group, the other two being $langle x^a,y^a rangle$ and $langle x^{a^2},y^{a^2}rangle$, where $x^a = a^{-1}xa$.
      $$langle a,b,x,y,mid a^3,b^2,(ab)^2,x^3,y^2,(xy)^2,
      [x,x^a],[y,x^a],[x,x^{a^2}],[y,x^{a^2}],
      [x,y^a],[y,y^a],[x,y^{a^2}],[y,y^{a^2}],
      [x,b], [y,b] rangle.$$
      A routine calculation from the presentation shows that $G/[G,G] cong C_2 times C_2$, but it is not hard to show that directly.



      PS: In fact the four relators $[x,x^{a^2}],[y,x^{a^2}],[x,y^{a^2}],[y,y^{a^2}]$ are redundant. The fact that $langle x,y rangle$ commutes with $langle x^{a^2},y^{a^2} rangle$ follows from the fact that $langle x,y rangle$ commutes with $langle x^{a},y^{a} rangle$ by conjugating by $b$.






      share|cite|improve this answer











      $endgroup$
















        3












        3








        3





        $begingroup$

        Here is a presentation of $S_3 wr S_3$. Note that $a,b$ generate the top $S_3$ factor, and $x,y$ generate one of the three factors of the base group, the other two being $langle x^a,y^a rangle$ and $langle x^{a^2},y^{a^2}rangle$, where $x^a = a^{-1}xa$.
        $$langle a,b,x,y,mid a^3,b^2,(ab)^2,x^3,y^2,(xy)^2,
        [x,x^a],[y,x^a],[x,x^{a^2}],[y,x^{a^2}],
        [x,y^a],[y,y^a],[x,y^{a^2}],[y,y^{a^2}],
        [x,b], [y,b] rangle.$$
        A routine calculation from the presentation shows that $G/[G,G] cong C_2 times C_2$, but it is not hard to show that directly.



        PS: In fact the four relators $[x,x^{a^2}],[y,x^{a^2}],[x,y^{a^2}],[y,y^{a^2}]$ are redundant. The fact that $langle x,y rangle$ commutes with $langle x^{a^2},y^{a^2} rangle$ follows from the fact that $langle x,y rangle$ commutes with $langle x^{a},y^{a} rangle$ by conjugating by $b$.






        share|cite|improve this answer











        $endgroup$



        Here is a presentation of $S_3 wr S_3$. Note that $a,b$ generate the top $S_3$ factor, and $x,y$ generate one of the three factors of the base group, the other two being $langle x^a,y^a rangle$ and $langle x^{a^2},y^{a^2}rangle$, where $x^a = a^{-1}xa$.
        $$langle a,b,x,y,mid a^3,b^2,(ab)^2,x^3,y^2,(xy)^2,
        [x,x^a],[y,x^a],[x,x^{a^2}],[y,x^{a^2}],
        [x,y^a],[y,y^a],[x,y^{a^2}],[y,y^{a^2}],
        [x,b], [y,b] rangle.$$
        A routine calculation from the presentation shows that $G/[G,G] cong C_2 times C_2$, but it is not hard to show that directly.



        PS: In fact the four relators $[x,x^{a^2}],[y,x^{a^2}],[x,y^{a^2}],[y,y^{a^2}]$ are redundant. The fact that $langle x,y rangle$ commutes with $langle x^{a^2},y^{a^2} rangle$ follows from the fact that $langle x,y rangle$ commutes with $langle x^{a},y^{a} rangle$ by conjugating by $b$.







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited Dec 1 '18 at 8:41

























        answered Nov 30 '18 at 8:12









        Derek HoltDerek Holt

        53.5k53571




        53.5k53571






























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