Presentation of wreath product $G=S_3 wr S_3$ of symmetric groups. What is the isomorphism type of $G/[G,G]$?
$begingroup$
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
$endgroup$
|
show 1 more comment
$begingroup$
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
$endgroup$
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
|
show 1 more comment
$begingroup$
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
$endgroup$
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
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
|
show 1 more comment
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
|
show 1 more comment
1 Answer
1
active
oldest
votes
$begingroup$
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 $b$.
$endgroup$
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
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
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2243607%2fpresentation-of-wreath-product-g-s-3-wr-s-3-of-symmetric-groups-what-is-the%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
$begingroup$
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 $b$.
$endgroup$
add a comment |
$begingroup$
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 $b$.
$endgroup$
add a comment |
$begingroup$
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 $b$.
$endgroup$
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 $b$.
edited Dec 1 '18 at 8:41
answered Nov 30 '18 at 8:12
Derek HoltDerek Holt
53.5k53571
53.5k53571
add a comment |
add a comment |
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.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2243607%2fpresentation-of-wreath-product-g-s-3-wr-s-3-of-symmetric-groups-what-is-the%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
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
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