Mixing
Show that $σ(i_1, i_2, . . . , i_k)σ^{−1} = (σ(i_1), σ(i_2), . . . , σ(i_k))$
$begingroup$
Here's the full question: If $σ ∈ S_n$ is any permutation and $i_1, . . . , i_k $ are $k$ distinct elements of ${1, . . . , n}$, show that $σ(i_1, i_2, . . . , i_k)σ^{−1} = (σ(i_1), σ(i_2), . . . , σ(i_k))$
I was given this hint:
For $j ∈ {1, . . . , n}$ consider these cases: (1) $j$ is one of the values $σ(i_1), . . . ,σ(i_{k−1})$, (2) $j = σ(i_k)$, (3) $j$ is not one of the values
$σ(i_1), . . . , σ(i_k)$.
I'm supposed to show that both sides send $j$ to the same thing, but the wording of this question confused me and the hint only worsened this for me.
permutations
edited Dec 19 '16 at 20:38
Hanul Jeon
17.7k42881
asked Apr 22 '16 at 2:08
ChrisChris
594510
$endgroup$
add a comment |
$begingroup$
Here's the full question: If $σ ∈ S_n$ is any permutation and $i_1, . . . , i_k $ are $k$ distinct elements of ${1, . . . , n}$, show that $σ(i_1, i_2, . . . , i_k)σ^{−1} = (σ(i_1), σ(i_2), . . . , σ(i_k))$
I was given this hint:
For $j ∈ {1, . . . , n}$ consider these cases: (1) $j$ is one of the values $σ(i_1), . . . ,σ(i_{k−1})$, (2) $j = σ(i_k)$, (3) $j$ is not one of the values
$σ(i_1), . . . , σ(i_k)$.
I'm supposed to show that both sides send $j$ to the same thing, but the wording of this question confused me and the hint only worsened this for me.
permutations
edited Dec 19 '16 at 20:38
Hanul Jeon
17.7k42881
asked Apr 22 '16 at 2:08
ChrisChris
594510
$endgroup$
add a comment |
$begingroup$
Here's the full question: If $σ ∈ S_n$ is any permutation and $i_1, . . . , i_k $ are $k$ distinct elements of ${1, . . . , n}$, show that $σ(i_1, i_2, . . . , i_k)σ^{−1} = (σ(i_1), σ(i_2), . . . , σ(i_k))$
I was given this hint:
For $j ∈ {1, . . . , n}$ consider these cases: (1) $j$ is one of the values $σ(i_1), . . . ,σ(i_{k−1})$, (2) $j = σ(i_k)$, (3) $j$ is not one of the values
$σ(i_1), . . . , σ(i_k)$.
I'm supposed to show that both sides send $j$ to the same thing, but the wording of this question confused me and the hint only worsened this for me.
permutations
edited Dec 19 '16 at 20:38
Hanul Jeon
17.7k42881
asked Apr 22 '16 at 2:08
ChrisChris
594510
$endgroup$
Here's the full question: If $σ ∈ S_n$ is any permutation and $i_1, . . . , i_k $ are $k$ distinct elements of ${1, . . . , n}$, show that $σ(i_1, i_2, . . . , i_k)σ^{−1} = (σ(i_1), σ(i_2), . . . , σ(i_k))$
I was given this hint:
For $j ∈ {1, . . . , n}$ consider these cases: (1) $j$ is one of the values $σ(i_1), . . . ,σ(i_{k−1})$, (2) $j = σ(i_k)$, (3) $j$ is not one of the values
$σ(i_1), . . . , σ(i_k)$.
I'm supposed to show that both sides send $j$ to the same thing, but the wording of this question confused me and the hint only worsened this for me.
permutations
permutations
edited Dec 19 '16 at 20:38
Hanul Jeon
17.7k42881
asked Apr 22 '16 at 2:08
ChrisChris
594510
edited Dec 19 '16 at 20:38
Hanul Jeon
17.7k42881
asked Apr 22 '16 at 2:08
ChrisChris
594510
edited Dec 19 '16 at 20:38
Hanul Jeon
17.7k42881
edited Dec 19 '16 at 20:38
Hanul Jeon
17.7k42881
edited Dec 19 '16 at 20:38
Hanul Jeon
17.7k42881
17.7k42881
asked Apr 22 '16 at 2:08
ChrisChris
594510
asked Apr 22 '16 at 2:08
ChrisChris
594510
asked Apr 22 '16 at 2:08
ChrisChris
594510
594510
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
See the solution to Exercise 5.17 (a) in my Notes on the combinatorial fundamentals of algebra, version of 10 January 2019. Note that your notation $left(i_1, i_2, ldots, i_kright)$ corresponds to my notation $operatorname{cyc}_{i_1,i_2,ldots,i_k}$.
At some point you'll get used to this kind of proof, and begin saying that they are obvious; right now, probably having all the details available is a good thing.
answered Dec 19 '16 at 21:07
darij grinbergdarij grinberg
11.5k33168
$endgroup$
add a comment |
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
});
}
});
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%2f1753613%2fshow-that-%25cf%2583i-1-i-2-i-k%25cf%2583%25e2%2588%25921-%25cf%2583i-1-%25cf%2583i-2-%25cf%2583i-k%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
$begingroup$
See the solution to Exercise 5.17 (a) in my Notes on the combinatorial fundamentals of algebra, version of 10 January 2019. Note that your notation $left(i_1, i_2, ldots, i_kright)$ corresponds to my notation $operatorname{cyc}_{i_1,i_2,ldots,i_k}$.
At some point you'll get used to this kind of proof, and begin saying that they are obvious; right now, probably having all the details available is a good thing.
answered Dec 19 '16 at 21:07
darij grinbergdarij grinberg
11.5k33168
$endgroup$
add a comment |
$begingroup$
See the solution to Exercise 5.17 (a) in my Notes on the combinatorial fundamentals of algebra, version of 10 January 2019. Note that your notation $left(i_1, i_2, ldots, i_kright)$ corresponds to my notation $operatorname{cyc}_{i_1,i_2,ldots,i_k}$.
At some point you'll get used to this kind of proof, and begin saying that they are obvious; right now, probably having all the details available is a good thing.
answered Dec 19 '16 at 21:07
darij grinbergdarij grinberg
11.5k33168
$endgroup$
add a comment |
$begingroup$
See the solution to Exercise 5.17 (a) in my Notes on the combinatorial fundamentals of algebra, version of 10 January 2019. Note that your notation $left(i_1, i_2, ldots, i_kright)$ corresponds to my notation $operatorname{cyc}_{i_1,i_2,ldots,i_k}$.
At some point you'll get used to this kind of proof, and begin saying that they are obvious; right now, probably having all the details available is a good thing.
answered Dec 19 '16 at 21:07
darij grinbergdarij grinberg
11.5k33168
$endgroup$
See the solution to Exercise 5.17 (a) in my Notes on the combinatorial fundamentals of algebra, version of 10 January 2019. Note that your notation $left(i_1, i_2, ldots, i_kright)$ corresponds to my notation $operatorname{cyc}_{i_1,i_2,ldots,i_k}$.
At some point you'll get used to this kind of proof, and begin saying that they are obvious; right now, probably having all the details available is a good thing.
answered Dec 19 '16 at 21:07
darij grinbergdarij grinberg
11.5k33168
edited Feb 3 at 6:27
answered Dec 19 '16 at 21:07
darij grinbergdarij grinberg
11.5k33168
answered Dec 19 '16 at 21:07
darij grinbergdarij grinberg
11.5k33168
answered Dec 19 '16 at 21:07
darij grinbergdarij grinberg
11.5k33168
11.5k33168
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%2f1753613%2fshow-that-%25cf%2583i-1-i-2-i-k%25cf%2583%25e2%2588%25921-%25cf%2583i-1-%25cf%2583i-2-%25cf%2583i-k%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
