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4 edited title

# AquestionarisingfromWhendo the representationtheorysizesofconjugacyclassesandsquaresofdegrees of irrepsgivethesamepartitionfora finite groupsgroup?

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I should admit the question below does not have a serious motivation. But still I found it somehow natural.

Let $G$ be a finite group of order $n$ with $h$ conjugacy classes. If $c_1,\ldots,c_h$ are the orders of the conjugacy classes of $G$, then clearly

$n=c_1+c_2+\ldots+c_h$.

Let now $\pi_1,\ldots,\pi_h$ be the pairwise non-isomorphic, irreducible complex representations of $G$, then it G$. It is well known that another partition of$n$of length$h$is given by the squares of the degrees$d_i$'s of the$\pi_i$'s:$n=d_1^2+d_2^2+\ldots+d_h^2$. Question: Assume that, up to reordering, the two partitions of$n$described above are the same. Then what can we say about$G$? Is$G$forced to be abelian? 2 added 16 characters in body I should admit the question below does not have a serious motivation. But still I found it somehow natural. Let$G$be a finite group of order$n$with$h$conjugacy classes. If$c_1,\ldots,c_h$are the orders of the conjugacy classes , of$G$, then clearly$n=c_1+c_2+\ldots+c_h$.If Let now$\pi_1,\ldots,\pi_h$are be the pairwise non-isomorphic, irreducible complex representations of$G$, then it is well known that another partition of$n$of length$h$is given by the squares of the degrees$d_i$'s of the$\pi_i$'s:$n=d_1^2+d_2^2+\ldots+d_h^2$. Question: Assume that, up to reordering, the two partitions of$n$described above are the same. Then what can we say about$G$? Is$G\$ forced to be abelian?

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