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Linear representations of algebras and groups, Lie theory, associative algebras, multilinear algebra.

5 votes
Accepted

Subgroups and representations of finite groups of Lie type

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 …
Nick Gill's user avatar
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2 votes

Common basis for permutation matrices

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 …
Nick Gill's user avatar
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20 votes
3 answers
934 views

What did Frobenius prove about $M_{12}$?

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 …
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4 votes

Choice of bilinear forms for simple groups

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 …
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5 votes
Accepted

Regular elementary abelian subgroups of primitive permutation groups

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 …
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3 votes
2 answers
235 views

Do the irreducible modules of this finite group preserve a tensor product structure?

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 …
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2 votes

Representations of orthogonal groups over the field of two elements

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 …
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8 votes
Accepted

Representations of $\text{SL}(n,\mathbb{F}_p)$ and $\text{Sp}(2n,\mathbb{F}_p)$ whose dimens...

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. …
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5 votes
Accepted

Special linear groups contained in symplectic groups

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 …
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17 votes
Accepted

Does the Alternating group of degree $n>7$ have exactly one irreducible character of degree ...

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 …
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4 votes

Largest permutation group without 2-cycles or 3-cycles

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 …
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9 votes

Maximal number of maximal subgroups

The document I linked to above is sufficiently striking as to warrant an answer of its own. I hope it complements the community wiki above. As mentioned above the relevant conjecture in this area is …
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3 votes
1 answer
127 views

Projective representations of extensions of $PSL_2(q)$

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\ …
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7 votes
Accepted

If all real conjugacy classes are strongly real, then all real irreps are "strongly real"(sy...

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 …
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4 votes

Subgroups of GL_2 over a finite field

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 …
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