### A characterization of characters which arise from ...-normalizers.

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Let $G$ be a finite group and $p$ a prime number. We prove that if $G$ is a finite group of order $\left|\mathrm{PSL}\right(2,{p}^{2}\left)\right|$ such that $G$ has an irreducible character of degree ${p}^{2}$ and we know that $G$ has no irreducible character $\theta $ such that $2p\mid \theta \left(1\right)$, then $G$ is isomorphic to $\mathrm{PSL}(2,{p}^{2})$. As a consequence of our result we prove that $\mathrm{PSL}(2,{p}^{2})$ is uniquely determined by the structure of its complex group algebra.

The notion of age of elements of complex linear groups was introduced by M. Reid and is of importance in algebraic geometry, in particular in the study of crepant resolutions and of quotients of Calabi–Yau varieties. In this paper, we solve a problem raised by J. Kollár and M. Larsen on the structure of finite irreducible linear groups generated by elements of age $\le 1$. More generally, we bound the dimension of finite irreducible linear groups generated by elements of bounded deviation. As a consequence...

In this article we prove an effective version of the classical Brauer’s Theorem for integer class functions on finite groups.