Unanswered Questions
2,207 questions with no upvoted or accepted answers
54
votes
0
answers
1k
views
Class function counting solutions of equation in finite group: when is it a virtual character?
Let $w=w(x_1,\dots,x_n)$ be a word in a free group of rank $n$. Let $G$ be a finite group. Then we may define a class function $f=f_w$ of $G$ by
$$ f_w(g) = |\{ (x_1,\dots, x_n)\in G^n\mid w(x_1,\dots,...
- 6,989
43
votes
0
answers
2k
views
Why are there so few quaternionic representations of simple groups?
Having spent many hours looking through the Atlas of Finite Simple Groups while in Grad school, I recall being rather intrigued by the fact that among the sporadic groups, only one (McLaughlin as I ...
- 11.5k
40
votes
0
answers
1k
views
Groups whose complex irreducible representations are finite dimensional
By a complex irreducible representation of a group $G$, I mean a simple $\mathbb CG$-module. So my representations need not be unitary and we are working in the purely algebraic setting.
It is easy ...
- 4,733
39
votes
0
answers
5k
views
Homology of $\mathrm{PGL}_2(F)$
Update: As mentioned below, the answer to the original question is a strong No. However, the case of $\pi_4$ remains, and actually I think that this one would follow from Suslin's conjecture on ...
- 21.8k
38
votes
0
answers
1k
views
Is this generalized character always a character?
Let $G$ be a finite group, and $p$ be a prime. Then there is a generalized character $\Psi$ of $G$ which takes value $0$ on all elements of order divisible by $p$, and has $\Psi(y)$ equal to the ...
- 45k
37
votes
0
answers
1k
views
Are there infinite versions of sporadic groups?
The classification of finite simple groups states roughly that every non-abelian finite simple group is either alternating, a group of Lie type, or a sporadic group.
For each of the groups of Lie ...
CommunityBot
- 1
34
votes
0
answers
2k
views
Next steps on formal proof of classification of finite simple groups
While people are steaming ahead on finessing the proof of the classification of finite simple groups (CFSG), we have a formal proof in Coq of one of the first major components: the Feit-Thompson odd-...
CommunityBot
- 1
34
votes
0
answers
1k
views
Is there a Mathieu groupoid M_31?
I have read something which said that the large amount of common structure between the simple groups $SL(3,3)$ and $M_{11}$ indicated to Conway the possibility that the Mathieu groupoid $M_{13}$ might ...
- 3,695
30
votes
0
answers
766
views
Do two integral matrices generate a free group?
Is it decidable whether two given elements of ${\rm GL}(n,{\mathbb Z})$ generate a free group of rank 2?
This is a simple question that I have been asking people for the past couple of years, but ...
- 38.1k
30
votes
0
answers
766
views
Cohomology of symmetric groups and the integers mod 12
When $n \ge 4$, the third homology group $H_3(S_n,\mathbb{Z})$ of the symmetric group $S_n$ contains $\mathbb{Z}_{12}$ as a summand. Using the universal coefficient theorem we get $\mathbb{Z}_{12}$ ...
- 23.5k
30
votes
0
answers
1k
views
Follow-up to Steinberg's problem (12) in his 1966 ICM talk?
Steinberg's lecture at the 1966 ICM in Moscow here surveyed his work on regular elements of semisimple algebraic groups, while also formulating a number of then-open questions as "problems" (...
- 36.7k
29
votes
0
answers
911
views
The field of fractions of the rational group algebra of a torsion free abelian group
Let $G$ be a torsion free abelian group (infinitely generated to get anything interesting). The group algebra $\mathbb{Q}[G]$ is an integral domain. Let $\mathbb{Q}(G)$ be its field of fractions.
...
CommunityBot
- 1
29
votes
0
answers
1k
views
Non-linear expanders?
Recall that a family of graphs (indexed by an infinite set, such as the primes, say) is called an expander family if there is a $\delta>0$ such that, on every graph in the family, the discrete ...
- 21.3k
28
votes
0
answers
692
views
Mathieu group $M_{23}$ as an algebraic group via additive polynomials
An elegant description of the Mathieu group $M_{23}$ is the following: Let $C$ be the multiplicative subgroup of order $23$ in the field $F=\mathbb F_{2^{11}}$ with $2^{11}$ elements. Then $M_{23}$ is ...
- 13.5k
27
votes
0
answers
961
views
A question on simultaneous conjugation of permutations
Given $a,b\in S_n$ such that their commutator has at least $n-4$ fixed points, is there an element $z\in S_n$ such that $a^z=a^{-1}$, and $b^z=b^{-1}$? Here $a^z:=z^{-1}az$.
Magma says that the ...
- 34.6k