Recent zbMATH articles in MSC 20C15 https://zbmath.org/atom/cc/20C15 2021-07-26T21:45:41.944397Z Werkzeug The structure of Grothendieck rings of dihedral groups https://zbmath.org/1463.19001 2021-07-26T21:45:41.944397Z "Tang, Shuai" https://zbmath.org/authors/?q=ai:tang.shuai Summary: The representation category of a dihedral group is a symmetric semisimple monoidal category, so the Grothendieck ring of such a category is a commutative ring generated by finitely many elements. In this paper, the minimal generators of the Grothendieck ring are determined. Moreover, it is shown that the Grothendieck ring is isomorphic to a quotient of a polynomial ring. A new characterization of $$A_p(2)$$ and $$A_{p-1}(2)$$ where $$2^p-1$$ is a prime https://zbmath.org/1463.20006 2021-07-26T21:45:41.944397Z "Babai, Azam" https://zbmath.org/authors/?q=ai:babai.azam "Khatami, Maryam" https://zbmath.org/authors/?q=ai:khatami.maryam Let $$G$$ be an arbitrary finite group. Then $$g\in G$$ is called a vanishing element if there exists an ordinary irreducible character $$\chi$$ of $$G$$ such that $$\chi(g)=0$$. The set of orders of all vanishing elements $$g\in G$$ is denoted by $$Vo(G)$$. Recently, it has been conjectured that if $$M$$ is a simple group with $$\vert M \vert = \vert G \vert$$ then $$Vo(M)=Vo(G)$$ implies that $$M\cong G$$ (see [\textit{M. Foroudi Ghasemabadi} et al., Sib. Math. J. 56, No. 1, 78--82 (2015; Zbl 1318.20012)]). The authors of the present paper prove the following special case of this conjecture. Recall that $$A_{k}(q)=\mathrm{PSL}(k,q)$$. (Main Theorem): Let $$p$$ be a Mersenne prime (so $$2^{p}-1$$ is also a prime). Then the conjecture holds whenever the simple group is $$M=A_{p}(2)$$ ($$p>2$$) or $$A_{p-1}(2)$$ ($$p>3$$). The idea of the proof is to show first that $$G$$ has a normal series of the form $$1\# H\# K\# G$$ where $$K/H$$ is a simple nonabelian group, and then use the hypotheses to show that $$K/H\cong M$$. The proof is given in detail for $$M=A_{p}(2)$$ but only sketched for the other case. The McKay conjecture for $$\pi$$-partial characters https://zbmath.org/1463.20007 2021-07-26T21:45:41.944397Z "Chang, Xuewu" https://zbmath.org/authors/?q=ai:chang.xuewu "Wang, Na" https://zbmath.org/authors/?q=ai:wang.na Summary: The McKay conjecture was studied for $$\pi$$-partial characters, and a canonical bijection was constructed between the two sets of corresponding irreducible $$\pi$$-partial characters based on a related theorem of Wolf. This result can be viewed as a $$\pi$$-theoretic version of McKay conjecture for monomial characters in which there is a canonical bijection. Inductive sources for $$\pi$$-partial characters https://zbmath.org/1463.20008 2021-07-26T21:45:41.944397Z "Jin, Ping" https://zbmath.org/authors/?q=ai:jin.ping "Mu, Jinqi" https://zbmath.org/authors/?q=ai:mu.jinqi Summary: The inductive sources in the theory of $$\pi$$-partial characters are discussed, and some properties of $${I_\pi}$$-inductive sources are given. By introducing the notions of $${B_\pi}$$-inductive sources and $${D_\pi}$$-inductive sources, a characterization of $${I_\pi}$$-inductive sources is obtained, and the relationship among these three inductive sources is described. The results strengthen Lewis' theorem, cover some of the classical results of inductive sources for complex characters, and also contain the corresponding theorems of inductive sources for Brauer characters. Nonsolvable groups with two nonlinear irreducible characters of $${p'}$$-degrees https://zbmath.org/1463.20009 2021-07-26T21:45:41.944397Z "Liu, Yang" https://zbmath.org/authors/?q=ai:liu.yang "Lu, Ziqun" https://zbmath.org/authors/?q=ai:lu.ziqun "Zhang, Jiping" https://zbmath.org/authors/?q=ai:zhang.jiping Summary: McKay conjecture is an important problem in representation theory of finite groups. In this paper, the authors consider nonsolvable group with two $${p'}$$-degree characters and prove the conjecture for such group. Furthermore, they show that if $$G$$ is a nonsolvable group such that $$|{\mathrm{Irr}}_1 (G, p')| = 2$$ for odd prime $$p$$, then $$p = 3$$, and $$G \cong {\mathrm{PSL}}_2 (7)$$ or there exists a normal 2-subgroup $$N$$ such that $$G/N \cong {\mathrm{PSL}}_2 (5)$$. On a maximal subgroup $$(2^9:(L_3(4)):3$$ of the automorphism group $$U_6(2):3$$ of $$U_6(2)$$ https://zbmath.org/1463.20010 2021-07-26T21:45:41.944397Z "Prins, Abraham Love" https://zbmath.org/authors/?q=ai:prins.abraham-love "Monaledi, Ramotjaki Lucky" https://zbmath.org/authors/?q=ai:monaledi.ramotjaki-lucky "Fray, Richard Llewellyn" https://zbmath.org/authors/?q=ai:llewellyn-fray.richard Summary: In this paper, the Fischer-Clifford matrices and associated character table of a maximal subgroup $$(2^9:(L_3(4)):3$$ of one of the automorphism groups $$U_6(2):3$$ of the unitary group $$U_6(2)$$ are constructed. Groups whose set of vanishing elements is exactly a conjugacy class https://zbmath.org/1463.20011 2021-07-26T21:45:41.944397Z "Robati, Sajjad Mahmood" https://zbmath.org/authors/?q=ai:robati.sajjad-mahmood Summary: Let $$G$$ be a finite group. We say that an element $$g$$ in $$G$$ is a vanishing element if there exists some irreducible character $$\chi$$ of $$G$$ such that $$\chi(g)=0$$. In this paper, we classify groups whose set of vanishing elements is exactly a conjugacy class.