Consistency with the Class Equation?
$begingroup$
Given a finite group of order $n$ and trivial center, the Class Equation reads:
begin{equation}
n=1+sum_{b_i in B}frac{n}{|C_G(b_i)|}
end{equation}
where $B$ is a set of representatives of the conjugacy classes. Under the same hypothesis, the following equation seems to hold (say $G=lbrace e,a_1,dots,a_{n-1} rbrace$):
begin{equation}
n(n-1)=sum_{j=1}^{n-1}delta(Lambda)_j|C_G(a_j)|
end{equation}
$delta(Lambda)$ being a $(n-1)$-term partition of the integer $Lambda:=sum_{substack{lambda_i in lambda(n-1)\ lambda_i>1}}lambda_i^2$, where $lambda(n-1)$ is a partition of the integer $n-1$. As my proof sketch reported here has not gotten a confirmation or a disproval, yet, I was wondering first of all whether such a linear combination were "clearly" contradicting the Class Equation, an event that would automatically disprove my proof.
Does anybody see any conflict between the two formulas? Of course, my best would be to have directly an answer to the submitted proof in the link above.
group-theory
asked Jan 15 at 22:43
LucaLuca
17919
$endgroup$
add a comment |
$begingroup$
Given a finite group of order $n$ and trivial center, the Class Equation reads:
begin{equation}
n=1+sum_{b_i in B}frac{n}{|C_G(b_i)|}
end{equation}
where $B$ is a set of representatives of the conjugacy classes. Under the same hypothesis, the following equation seems to hold (say $G=lbrace e,a_1,dots,a_{n-1} rbrace$):
begin{equation}
n(n-1)=sum_{j=1}^{n-1}delta(Lambda)_j|C_G(a_j)|
end{equation}
$delta(Lambda)$ being a $(n-1)$-term partition of the integer $Lambda:=sum_{substack{lambda_i in lambda(n-1)\ lambda_i>1}}lambda_i^2$, where $lambda(n-1)$ is a partition of the integer $n-1$. As my proof sketch reported here has not gotten a confirmation or a disproval, yet, I was wondering first of all whether such a linear combination were "clearly" contradicting the Class Equation, an event that would automatically disprove my proof.
Does anybody see any conflict between the two formulas? Of course, my best would be to have directly an answer to the submitted proof in the link above.
group-theory
asked Jan 15 at 22:43
LucaLuca
17919
$endgroup$
add a comment |
$begingroup$
Given a finite group of order $n$ and trivial center, the Class Equation reads:
begin{equation}
n=1+sum_{b_i in B}frac{n}{|C_G(b_i)|}
end{equation}
where $B$ is a set of representatives of the conjugacy classes. Under the same hypothesis, the following equation seems to hold (say $G=lbrace e,a_1,dots,a_{n-1} rbrace$):
begin{equation}
n(n-1)=sum_{j=1}^{n-1}delta(Lambda)_j|C_G(a_j)|
end{equation}
$delta(Lambda)$ being a $(n-1)$-term partition of the integer $Lambda:=sum_{substack{lambda_i in lambda(n-1)\ lambda_i>1}}lambda_i^2$, where $lambda(n-1)$ is a partition of the integer $n-1$. As my proof sketch reported here has not gotten a confirmation or a disproval, yet, I was wondering first of all whether such a linear combination were "clearly" contradicting the Class Equation, an event that would automatically disprove my proof.
Does anybody see any conflict between the two formulas? Of course, my best would be to have directly an answer to the submitted proof in the link above.
group-theory
asked Jan 15 at 22:43
LucaLuca
17919
$endgroup$
Given a finite group of order $n$ and trivial center, the Class Equation reads:
begin{equation}
n=1+sum_{b_i in B}frac{n}{|C_G(b_i)|}
end{equation}
where $B$ is a set of representatives of the conjugacy classes. Under the same hypothesis, the following equation seems to hold (say $G=lbrace e,a_1,dots,a_{n-1} rbrace$):
begin{equation}
n(n-1)=sum_{j=1}^{n-1}delta(Lambda)_j|C_G(a_j)|
end{equation}
$delta(Lambda)$ being a $(n-1)$-term partition of the integer $Lambda:=sum_{substack{lambda_i in lambda(n-1)\ lambda_i>1}}lambda_i^2$, where $lambda(n-1)$ is a partition of the integer $n-1$. As my proof sketch reported here has not gotten a confirmation or a disproval, yet, I was wondering first of all whether such a linear combination were "clearly" contradicting the Class Equation, an event that would automatically disprove my proof.
Does anybody see any conflict between the two formulas? Of course, my best would be to have directly an answer to the submitted proof in the link above.
group-theory
group-theory
asked Jan 15 at 22:43
LucaLuca
17919
asked Jan 15 at 22:43
LucaLuca
17919
edited Jan 16 at 22:22
Luca
asked Jan 15 at 22:43
LucaLuca
17919
asked Jan 15 at 22:43
LucaLuca
17919
asked Jan 15 at 22:43
LucaLuca
17919
17919
add a comment |
add a comment |
0
active
oldest
votes
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%2f3075073%2fconsistency-with-the-class-equation%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
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%2f3075073%2fconsistency-with-the-class-equation%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
