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It seems that the answer is yes. A MathSciNet search brought up the paper Y. Berkovich, D. Chillag, and M. Herzog, Finite groups in which the degrees of the nonlinear irreducible characters are di …

Here's another proof. If $G$ acts transitively on $X$, then the permutation representation of $G$ is induced from the permutation representation of the stabilizer $S$ of an arbitrary point. As a resu …

Yes. If $f = (\mathcal O : R)$, then $f\mathcal O \subset R$ and consequently $\text{SL}_n(R)$ contains $\ker (\text{SL}_n(\mathcal O) \to \text{SL}_n(\mathcal O/f\mathcal O))$, which has finite index …

The answer is no: see Corollary 1.3 in Robert M. Guralnick; Gunter Malle; Gabriel Navarro, Self-normalizing Sylow subgroups, Proc. Amer. Math. Soc. 132 (2004), 973-979.

Here are three examples. A finite field extension of $\mathbb R$ must be quadratic. For if $\mathbb R^n$ carries the structure of a field, then its group of units $\mathbb R^n - \{0\}$ is an abelian …

Corollary 7.54 in Hoffman and Morris, The Structure of Compact Groups, seems to be what you want.

I know of two clean approaches to classifying the unitary irreps of the $ax+b$ group. The first is to write the group as a semidirect product $\mathbb R \ltimes \mathbb R_{>0}$. There is a theory (due …

We can also rule out the case of commutative $R$ by appealing to the Artin–Procesi theorem: an Azumaya algebra of constant rank $(n+1)^2$ (e.g. $M_{n+1}(R)$) satisfies all the $\mathbb Z$-multilinear …

If $A$ is the character table and $A^\ast$ is its conjugate transpose, then the orthogonality relations tell us that $A A^\ast = \text{diag}\{|C_G(g)|\} $, where the enties run over a fixed choice of …

The answer to your first question is negative. For a concrete example, you can show that the conjugacy class rings of the nonisomorphic groups $Q_8$ and $D_8$ are isomorphic, via an isomorphism that p …

I believe the answer is yes. Let's begin by recalling that if one wants to show that a locally compact group $G$ is of type I, it suffices to show that $G$ contains a "large" compact subgroup $K$, in …