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Let $G$ be a finite group, let $F$ be a finite field and let $F[G]$ be the group algebra of $G$ over $F$.

  1. What is known about the structure of the group of units $F[G]^\times$? Of course, it must contain a subgroup isomorphic to $G$.

  2. Is there anything special about how $G$ is embedded in $F[G]^\times$?

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    $\begingroup$ mathoverflow.net/questions/142899/units-in-a-group-algebra To put it simply: for your first question, answer is "something is known, but not much". For your second one — I'd say that finding the right way to ask it would be a major breakthrough in modular representation theory. $\endgroup$
    – Denis T
    Commented Jun 20, 2023 at 4:25

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