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We prove that the number of conjugacy classes of a finite groupGconsisting of elements of odd order, is larger than or equal to that number for the normaliser of a Sylow 2-subgroup ofG. This is predicted by the Alperin Weight Conjecture.

 

We completely describe all the possible fields of values of irreducible characters of degree up to 3 of finite groups. The obtained result points toward a rather surprising connection between the field of values and the degree of an arbitrary irreducible character. 

We prove that if G is a nonsolvable group, then the proportion of vanishing elements of G is at least 1067/1260 (and this lower bound is optimal). This confirms a conjecture of Dolfi, Pacifici, and Sanus [7]. 

Abstract For every integer k there exists a bound $$B=B(k)$$ B = B ( k ) such that if the characteristic polynomial of $$g\in \textrm{SL}_n(q)$$ g ∈ SL n ( q ) is the product of $$\le k$$ ≤ k pairwise distinct monic irreducible polynomials over $$\mathbb {F}_q$$ F q , then every element x of $$\textrm{SL}_n(q)$$ SL n ( q ) of support at least B is the product of two conjugates of g . We prove this and analogous results for the other classical groups over finite fields; in the orthogonal and symplectic cases, the result is slightly weaker. With finitely many exceptions ( p ,  q ), in the special case that $$n=p$$ n = p is prime, if g has order $$\frac{q^p-1}{q-1}$$ q p - 1 q - 1 , then every non-scalar element $$x \in \textrm{SL}_p(q)$$ x ∈ SL p ( q ) is the product of two conjugates of g . The proofs use the Frobenius formula together with upper bounds for values of unipotent and quadratic unipotent characters in finite classical groups. 

The deep theory of approximate subgroups establishes three-step product growth for subsets of finite simple groups $G$ of Lie type of bounded rank. In this paper, we obtain two-step growth results for representations of such groups $G$ (including those of unbounded rank), where products of subsets are replaced by tensor products of representations. Let $G$ be a finite simple group of Lie type and $\chi $ a character of $G$. Let $|\chi |$ denote the sum of the squares of the degrees of all (distinct) irreducible characters of $G$ that are constituents of $\chi $. We show that for all $\delta>0$, there exists $\epsilon>0$, independent of $G$, such that if $\chi $ is an irreducible character of $G$ satisfying $|\chi | \le |G|^{1-\delta }$, then $|\chi ^2| \ge |\chi |^{1+\epsilon }$. We also obtain results for reducible characters and establish faster growth in the case where $|\chi | \le |G|^{\delta }$. In another direction, we explore covering phenomena, namely situations where every irreducible character of $G$ occurs as a constituent of certain products of characters. For example, we prove that if $|\chi _1| \cdots |\chi _m|$ is a high enough power of $|G|$, then every irreducible character of $G$ appears in $\chi _1\cdots \chi _m$. Finally, we obtain growth results for compact semisimple Lie groups.

 

Let be a finite group and and be different primes. Assume that is odd and . We prove that if divides the degrees of the nonlinear irreducible ‐modular representations, then has a normal ‐complement.

 

Abstract We prove that any element in a sufficiently large transitive finite simple permutation group is a product of two derangements.