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A full classification of such representations (and much more) can be found here: Prime power degree representations of quasi-simple groups by Malle and Zalesskii You can read this paper here. …

Let $K=PSL_2(q)$ where $q=p^a$ for some odd prime $p$, and let $G$ be a group such that $G/O(G)\cong K$. (Here $O(G)$ is the largest odd-order normal subgroup of $G$.) I have a homomorphism $\phi: G\ …

I presume you're talking about characters over C. In which case Green's classic paper "The characters of the finite general linear groups" is the only thing I know on this subject. He defines a whole …

Old answer: You know already that the answer is ``yes.'' For a reference, see result 2 of Rasala, Richard On the minimal degrees of characters of $S_n$. J. Algebra 45 (1977), no. 1, 132–181. Thi …

You can find a nice, concise introduction in $\S5.4$ of Kleidman and Liebeck's The subgroup structure of the finite classical groups. They focus on the modular theory over $\overline{\mathbb{F}_2}$, b …

I think the key word you need here is reflexive. A bilinear form $\beta$ is reflexive if $$\beta(x,y)=0 \Longrightarrow \beta(y,x)=0.$$ It's pretty clear that any sensible notion of orthogonality (and …

Dickson is responsible for the classification of subgroups of $SL_2(\mathbb{F}_q)$ (and once you've got this the subgroups of $GL_2(\mathbb{F}_q)$ are easy). You can find a full proof in Suzuki's "Gro …

Explicit values for the minimum degree of a primitive permutation representation of a simple group of Lie type can be found in Table 4 of this paper: Guest, Simon; Morris, Joy; Praeger, Cheryl E.; Spi …

The question is this: When can $SL(m,q^k)$ be a subgroup of $Sp(2n,q)$ with $mk=2n$? As you point out, this is possible if $m=2$. There are many particular cases that can be ruled out by order c …

This is really an comment to @Dima's answer, but it's a bit long... There is a classical result of Jordan in permutation group theory that says the following: If a primitive group $G$ [on a set o …

This is a bit long for a comment. To answer the question it would be very useful to know what groups can act primitively on a set of order $2^a$. Looking at O'Nan--Scott--Aschbacher, we see that such …

The answer is No. My source is Rod Gow's rather wonderful little paper "Real-valued characters and the Schur index" (MSN). Let me quote from the introduction: It may happen that all elements of a …

I am interested in a particular group $G$, where $$ (A_4\times C_\ell) \lhd G \lhd S_4 \times D_\ell$$ Here, $C_\ell$ is cyclic, $D_\ell$ is dihedral of order $2\ell$, and the two inclusions both have …

I am interested in this paper which I can't read because it's in German: Frobenius, G., Über die Charaktere der mehrfach transitiven Gruppen., Berl. Ber. 1904, 558-571 (1904). ZBL35.0154.02. A free …

Here's a possible method: We might as well assume that $A$ is already a permutation matrix -- just conjugate $A$ and $B$ by the same unitary matrix $C$ and relabel. Now if we are looking to conjugat …