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I am reading a paper written by Masao Kiyota and Tetsuro Okuyama on "A Note on a Conjecture of K. Harada" and my question is regarding a statement in this paper. A link to the paper is provided at the end of the question.

Notation: Let $G$ be a finite group and $p$ be a prime number. Let $\{\chi_1,...,\chi_s\}$ be the set of all irreducible complex characters of $G.$ For a subset $J$ of the index set $\{1,...,s\}$, we put $\{\chi_J\}=\{\chi_j:j\in J\}$ and $\rho_J=\sum_{j\in J} \chi_j(1) \chi.$

Harada's Conjecture: If $\rho_J(x)=0$ for any $p-$ singular element $x$ of $G,$ then $\{\chi_J\}$ is a union of $p-$ blocks of $G.$

Harada's Conjecture(A): If $\rho_J(x)=0$ for any $p-$ singular element $x$ of $G$ and $\{\chi_J\}\subset B$ for a $p-$ block $B$ of $G,$ then $\{\chi_J\}=\emptyset$ or $B.$

My Question: The paper I mentioned earlier implies that Harada's conjecture is equivalent to Harada's conjecture(A). This, apparently, has been proved by Harada himself but I am unable to find his proof. Could someone please explain to me how Harada's original version can be reduced to version(A)?

My background: I have read up to chapter 7 of Martin Isaacs' Character Theory book. Coming summer, I am planning on reading Gabriel Navarro's Characters and Blocks of Finite Groups with a professor, so I want to read some fun papers and get some idea about the subject. I binge read first couple of chapters of Navarro's book.

Thank you!!

Paper: https://projecteuclid.org/download/pdf_1/euclid.pja/1195516542

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  • $\begingroup$ What's a $p$-singular element, and what's a $p$-block? Also your link is broken. $\endgroup$
    – anon
    Commented Apr 7, 2021 at 10:46
  • $\begingroup$ @runway44 If an element has order divisible by p it's called p-singular (otherwise it's p-singular). The definition of a p-block requires some ground work (I believe) so I will write an update to the question as soon as possible. Also I just realized the link is broken. I have fixed it now. Thank you for your interest in the problem! $\endgroup$
    – user911589
    Commented Apr 7, 2021 at 18:39
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    $\begingroup$ It's Section 2 of Harada's paper, entitled "2. Reduction from Conjecture A to Conjecture A' " The paper is J. Algebra 70 (1981), 350--355. $\endgroup$
    – David A. Craven
    Commented Apr 7, 2021 at 20:19
  • $\begingroup$ @DavidA.Craven Thank you so much! I tried to find this paper by name earlier, but I couldn't. You made my life very easy by saying where exactly to look. $\endgroup$
    – user911589
    Commented Apr 7, 2021 at 23:22

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