Mixing
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
22419
$endgroup$
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
22419
$endgroup$
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
22419
$endgroup$
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
22419
asked Jan 15 at 22:43
LucaLuca
22419
edited Jan 16 at 22:22
Luca
asked Jan 15 at 22:43
LucaLuca
22419
asked Jan 15 at 22:43
LucaLuca
22419
asked Jan 15 at 22:43
LucaLuca
22419
22419
add a comment |
add a comment |
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