Elden Ring: Deluxe Edition Product Key [+ DLC]Torrent
Name Elden Ring
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Format File
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________________________________________
Elder Scrolls Online® is a registered trademark of ZeniMax Online Studios LLC.
Copyright © 2016 ZeniMax Online Studios LLC.
All other trademarks are the property of their respective owners.Q:

A non-isomorphism between simple groups

Let $\mathbb{F}_p$ be a finite field of characteristic $p$. If $G$ is a simple group and $\chi \colon G \rightarrow \mathbb{C}^*$ is a character, is it possible that $\chi$ does not lift to an irreducible representation of $G$? If not, is it the case that for a fixed prime $p$, there are only finitely many such simple groups $G$?

A:

I think the answer is yes: by the Lang–Stabilization theorem, for fixed $p$, there are finitely many isomorphism classes of simple groups of Lie type over $\mathbf{F}_p$. So suppose $G$ is a simple group of Lie type over $\mathbf{F}_p$, and suppose $\chi$ is a character of $G$ which does not lift to an irreducible representation. For fixed $p$, there are only finitely many simple groups of Lie type over $\mathbf{F}_p$, so there is a positive integer $n$ such that $G$ has a irreducible character of degree $n$. Since $G$ is simple and $\chi$ does not lift to an irreducible representation, there is an irreducible character $\varphi$ of $G$ such that $\varphi
ot\simeq \chi$. Write $\chi$ as a linear combination of fundamental characters: $|G|\,\chi = a_1\varphi_1 + a_2\varphi_2 + \dotsb + a_k\varphi_k$, where $a_1,\dotsc,a_k \in \mathbf{Z}_{\ge 0}$ and $a_1+\dotsb+a_k
e 0$ (as $G$ is simple, $k \ge 2$). Then $k \le 1$: if $k > 1$, then $\chi$ and $\varphi$ would be in the same irreducible representation

 

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    Python regex preg_replace() issue

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    import pygame
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