Geodesic
On the number of Heisenberg characters of finite groups
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Algebra, Tome 177 (2020), pp. 24-33

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An irreducible character $\chi$ of a finite group $G$ is called a Heisenberg character if $\ker \chi \supseteq [G, [G, G]]$. In this paper, we prove that the group $G$ has exactly $r$, $r \leq 3$, Heisenberg characters if and only if $|{G}/{G'}|=r$. If $G$ has exactly four Heisenberg characters, then $|{G}/{G'}|=4$, but the converse is not correct in general. Finally, it is proved that if $G$ has exactly five Heisenberg characters, then $|{G}/{G'}|=5$ or $|{G}/{G'}|=4$ and one of the Heisenberg characters of $G$ has the degree $2$.
Keywords: irreducible character, Heisenberg character, finite group. 
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     author = {A. Zolfi and A. R. Ashrafi},
     title = {On the number of {Heisenberg} characters of finite groups},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {24--33},
     publisher = {mathdoc},
     volume = {177},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTO_2020_177_a2/}
}
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A. Zolfi; A. R. Ashrafi. On the number of Heisenberg characters of finite groups. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Algebra, Tome 177 (2020), pp. 24-33. http://geodesic.mathdoc.fr/item/INTO_2020_177_a2/