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Questions tagged [finite-groups]

Questions on group theory which concern finite groups.

10
votes
2answers
321 views

Order of unipotent matrices over $\mathbb{Z}/q\mathbb{Z}$

Let $q$ be a prime power and let $n\geq2$ be an integer. Is it known what is the largest order of a unipotent upper-triangular $n\times n$ matrix over the ring $\mathbb{Z}/q\mathbb{Z}$? I am mostly ...
0
votes
0answers
121 views

Finite group and cyclic cover

Suppose the finite group $N$ surjects to finite group $F$. It is true that for any $G = N ⋊_α \mathbb{Z}$ there are infinitely many covers of $G$ that are cyclic and surject to $F$. But is this ...
-1
votes
1answer
204 views

Group isomorphism $R^{\times} \simeq C_n\times C_2$? [closed]

Let $n$ be an positive integer and $R:=\mathbb{Z}[X]/(X^2,nX)$. Do we have $R^{\times} \simeq C_n\times C_2$ (the group of units of $R$) as it seems to be the case for small values of $n$. If so, do ...
19
votes
3answers
663 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 ...
2
votes
1answer
64 views

Maximal subgroups of some special cases of ${\rm L}_{n}^{\epsilon}(q)$

Let $G$ be a finite simple group of type ${\rm L}_{n}^{\epsilon}(q)$ with the following conditions: $n$ and $\dfrac{q^{n}-\epsilon}{(q-\epsilon)(n,q-\epsilon)}$ both prime, $n\geqslant3$ and $(n,q,\...
0
votes
1answer
326 views

Is there any number other than 109 whose reciprocal contains the Fibonacci sequence? [closed]

Let $p$ be any odd number, and compute $1/p$ to $p$ decimal places. Compare your answer with the string that is formed by appending all remainders of $(10^n\ \text{mod p}) \text{ mod p}$ where ${0 <...
2
votes
0answers
155 views

Random walk on a finite group, converging modulo a function

Let $G$ be a finite group, and let $Q$ be a probability measure on $G$. Suppose that $Q$, as a function on $G$, is supported on a conjugacy class $C$. We denote by $Q^{*k}$ the $k$-fold convolution of ...
13
votes
4answers
414 views

Non-split Aut(G) $\to$ Out(G)?

I'm looking for examples of outer automorphisms of a finite group $G$ which do not lift to automorphisms (i.e. non-split quotient map $\mathrm{Aut}(G)\to \mathrm{Out}(G)$, where $\mathrm{Out}(G) = \...
2
votes
0answers
51 views

Number of orthogonal operators in representations of the Unitary Group

Let $G={\rm SU}(d)$ be the unitary group and $\rho(g)$ an irreducible representation of $g\in G$ in a $D$ dimensional Hilbert space $V$. Let $e_i\in V$ be the diagonal matrix whose only non-zero ...
1
vote
1answer
144 views

Is there an elementary reason for why $SL_2(\mathbb{F}_p)$ for $p>5$ does not embed into $SL_2(\mathbb{Z}_p[w])?$

This is an exercise from Serre's book on Galois cohomology. Let $p>5$ and consider the groups $SL_2(\mathbb{F}_p)$ and $SL_2(\mathbb{Z}_p[w])$ where $w$ is a primitive $p$th root of unity. Is ...
4
votes
1answer
165 views

No lifts in an exact sequence of profinite groups?

In pg. 24 of his book on Galois cohomology, Serre gives the following exercise: "Give an example of an extension $1 \to P \to E \to G \to 1$ of profinite groups with the following properties: (i) $...
1
vote
0answers
36 views

Alternating Hurwitz quotients multiplicity

How many times is each alternating group a hurwitz quotient? In other words, how many non-isomorphic ways can an alternating group be generated by an element of order 2 and an element of order 3 whose ...
2
votes
0answers
259 views

Symplectic group over finite field and quaternions

Symplectic group over finite field is defined as group preserving non-degenerate antisymmetric bilinear form on $\mathbb F_q^{2n}$. How could we define this group using quaternions ? This should be ...
2
votes
0answers
63 views

Is there any nice way to compute transfer homomorphism in a $p$-group?

Let $G$ be a $p$-group and $H$ be a subgroup of $G$. Set $V$ be a pretransfer map from $G$ to $H$. That is, $$V(g)=\prod_{t\in T} tg(t.g)^{-1}=\prod_{t\in T_0}tg^n_tt^{-1} $$ Can we say that the ...
3
votes
3answers
290 views

Computing the inner automorphism group of a finite Lie algebra

I'm interested in writing GAP code to compute the inner automorphism group of a finite Lie algebra. (I'd like to be able to group conjugate subalgebras together.) I've had trouble finding good ...
2
votes
0answers
231 views

Union of the conjugates of maximal subgroups

This post is a generalization of Union of the conjugates of a proper subgroup. Consider an interval $[H,G]$ in the subgroup lattice of the finite group $G$, with $H \neq G$ and such that: (1) $ \...
14
votes
1answer
411 views

The number of involutions in a permutation group

If $G$ is a group let $I(G)$ be the number of involutions (elements of order 2) in $G$. My question is then easily stated: does there exists a constant $C > 1$ such that for every $n \ge 1$ and ...
2
votes
0answers
122 views

Multiplicative subgroups of $GL(V)$ which are almost additively closed

Edit: According to comments of YCor and Vincent, I revise the question.I appreciate their comments: Let $G$ be a group. We say that $G$ is a special group if it is isomorphic to a subgroup $H$ ...
6
votes
0answers
179 views

Geometric interpretation of a formula for the induced character (fix point localization?)

Let $H < G$ be a subgroup of a finite group $G$. Let $X:=G/H$ and $\mathcal{F} \in Sh_G(X)$ be an equivariant sheaf on $X$ (w.r.t. left multiplication) associated to a finite dimensional ...
5
votes
2answers
221 views

Counting transitive generators according to coset type

Let $\sigma=(1\;2)(3\;4)\cdots (n-1\; n)$ be a fixed-point-free involution in $S_{2n}$. I want to count permutations $\pi$ such that the group $\langle \pi,\sigma\rangle$ generated by $\pi$ and $\...
4
votes
1answer
138 views

Are descents in alternating subgroup counted by $h$-vector?

Consider the alternating subgroup $A_n$ of the symmetric group $S_n$ (or in general any Coxeter Group). Is there a simplicial complex whose $h$-vector $h_i$ equals the number of elements of $A_n$ with ...
4
votes
0answers
114 views

subgroups of $\mathrm{Sp}_{2g}(\mathbb{Z}_2)$ whose mod-2 image is the symmetric group

Let $G \subseteq \mathrm{Sp}_{2g}(\mathbb{Z}_2)$ be a closed subgroup of the symplectic group over the $2$-adic integers whose image under the mod-$2$ homomorphism $\pi : \mathrm{Sp}_{2g}(\mathbb{Z}_2)...
1
vote
2answers
279 views

What are all the transitive extensions of cyclic groups?

"Let $G$ be a transitive group of permutations on a given set of letters. Let a new fixed letter be adjoined to every permutation of $G$. Then a transitive group $H$ of permutations on the combined ...
2
votes
1answer
111 views

Realization of the primitive action of a wreath product in a Galois group

Let $f$ be a polynomial over a field $K$ of degree $n$ such that $f(x^2)$ is separable. Assume that the Galois group $G$ of (a splitting field of ) $f(x^2)$ is maximal, that is to say, the wreath ...
2
votes
0answers
160 views

Sum of reciprocals in finite fields

Let $p$ be an odd prime number which large enough. I am interested in the study of the sums of reciprocals in the field $\mathbb{F}_p$. In particular, I have the following question: which primes $p$ ...
10
votes
1answer
314 views

Maximal subgroups of simple groups with normal $2$-subgroups

Let $G$ be a finite non-abelian simple group. Question: Does there always exist a maximal subgroup $M$ of $G$ such that $M$ has a non-trivial normal elementary abelian $2$-subgroup? This seems ...
6
votes
2answers
227 views

Finite lattice representation problem checking

[Grätzer and Schmidt 1963] proves that every algebraic lattice is isomorphic to the congruence lattice of a universal algebra. A finite lattice is algebraic. The finite lattice representation problem ...
5
votes
1answer
387 views

Is the Normal centralizer problem in P?

Notation $\le$ is used for the subgroup relation; $P$ means polynomial time in input size; $\Omega = \{1,2,3,\cdots,n\}$ is a input domain; $\mathrm{Sym}(\Omega)$ means the symmetric group on $\...
2
votes
0answers
286 views

Finite simple subgroups of $\mathrm{GL}_n(\mathbb{C})$

Let $n$ be a fixed positive integer. Is it true that there are only finitely many isomorphism classes of finite nonabelian simple subgroups of $\mathrm{GL}_n(\mathbb{C})$? I'm especially interested ...
2
votes
1answer
153 views

Recognition of finite simple groups by number of Sylow p-subgroups (2)

This is a follow-up to this question. Let $G$ and $G'$ be two finite simple groups of the following structures: 1- $A_{p}$, for some primes $p$; 2- $PSL_{p}(q)$, for some prime $p$ and some prime ...
7
votes
2answers
261 views

Reference request: Models of cuspidal representations of GL(n,k) where k is a finite field

Let $k=\mathbb{F}_q$ where $q$ is a prime power of odd cardinality. Where could I find explicit models of all irreducible cuspidal (complex) representations of $GL_n(k)$ for $n\ge 3$? I understand ...
10
votes
3answers
414 views

Low-dimensional irreducible 2-modular representations of the symmetric group

I apologize if this question is a little too basic for MathOverflow, but it's somewhat outside of my background and I'm frustrated that the answer doesn't seem to be explicit in the literature even ...
5
votes
1answer
135 views

Maximal subgroups of a finite simple group which have some minimal subgroups in common

Let $G$ be a finite simple group and $M$ and $M'$ be two maximal subgroups of $G$. Also let $m_{M}$ be the set of minimal subgroups of $M$ and similarly $m_{M'}$ be the set of minimal subgroups of $M'$...
0
votes
1answer
131 views

Recognition of finite simple groups by number of Sylow p-subgroups

Let $G$ and $G'$ be two finite simple groups and $p$ be a prime divisor of $\vert G\vert$ and $\vert G'\vert$. Also suppose that every Sylow p-subgroup of $G$ and $G'$ is a prime order subgroup($C_{p}$...
8
votes
3answers
381 views

Do actions of BS(1,n) on finite sets factor through abelian quotients?

Suppose $BS(1,n)$ is the Baumslag-Solitar group and $S_m$ is the symmetric group. If $\Phi: BS(1,n) \to S_m$ is a homomorphism, must the image of $\Phi$ be abelian?
4
votes
2answers
202 views

Ext in symmetric algebras and group algebras

Let $A$ be a selfinjective algebra and for an indecomposable module $M$ define $\psi_M:= \inf \{ i \geq 1 | Ext_A^i(M,M) \neq 0 \}$. Questions: In case $A$ is symmetric, do we have $\psi_M \leq max \...
8
votes
1answer
343 views

This group is “dual” to the Mathieu group $M_{23}$. Is it known?

Inspired by this question, in particular by the indeed elegant description of the Mathieu group $M_{23}$ it starts with, I am wondering about the following: Instead of $C$, defined as the ...
1
vote
1answer
195 views

About Frobenius Determinant Theorem

Finite group $G=\{x_1,x_2,...x_n\}$. Consider $G$'s multiplication table to be an $n\times n$ matrix $A$. Set $x_i=1$, $x_j=0$ ($j≠i$), $1≤i≤n$, then we get $n$ permutation matrices $S_i$ ($1≤i≤n$) s....
3
votes
0answers
144 views

a universal module for group cohomology?

I noticed the following funny fact when studying cohomology of finite groups. I explain it in the case of $H^2$ but it generalizes. Consider a two-cocycle $a\in Z^2(G,A)$ where $A$ is a left G-module....
2
votes
0answers
77 views

What is the importance and Application of studying degree of commutativity in finite group

What are the application and importance (or significance) of studying commutativity degree in a finite group $G$. i.e Significance of finding probability that arbitrary chosen element of the group ...
4
votes
1answer
147 views

Splitting of central extension read on 2-Sylow?

Let $G$ be a finite group with a central subgroup $Z$ of order 2. Suppose that $Z$ has a direct factor in some (and hence any) 2-Sylow subgroup of $G$. Does this imply that $Z$ has a direct factor in $...
1
vote
1answer
188 views

Extensions of a projective special linear group

Is it possible to classify groups that can be decomposed as a semidirect product $G=\mathrm{PSL}_2(q) \rtimes \langle t\rangle$, such that $t\in G$ has order $4$ and $Z(G)=\langle t^2\rangle$?
3
votes
1answer
140 views

Intersections of products of Sylow $p$-subgroups

Motivated by the work of Cohn, Umans and collaborators on bounding the complexity of matrix multiplication, a student and I have been investigating the following problem. For subsets $X$ and $Y$ of a ...
-2
votes
1answer
137 views

Number of subgroups of a group of orders $p^3$ [closed]

Let $p$ be a prime number. Is there a formula for the number of subgroups of $$\mathbb{Z}/p\mathbb{Z}\times \mathbb{Z}/p^2\mathbb{Z}$$ $$\mathbb{Z}/p\mathbb{Z}\times \mathbb{Z}/p\mathbb{Z}\times\...
4
votes
1answer
194 views

How to explicitly expand a class function in terms of irreducible characters?

Let $G$ be a finite group of exponent $n$ and let $d\mid n$. Consider the class function $$ f(g) = \begin{cases} 1 & g^d =1\\0&\textrm{otherwise}. \end{cases} $$ As a class function $f$ can ...
14
votes
0answers
614 views

Algebra for the Baby

I am reading the following article. Ryba, Alexander J.E., A natural invariant algebra for the Baby Monster group., J. Group Theory 10, No. 1, 55-69 (2007). ZBL1228.20012.. Author works with 4370-...
1
vote
1answer
82 views

Number of generators for the Schur multiplier of a finite group

Let $G$ be a finite group, and let $M(G)=H_2(G,\mathbb{Z})$ be its Schur multiplier. Are there any known bounds on the number of generators of $M(G)$ in terms of $G$? For example, if $G$ is abelian of ...
3
votes
0answers
144 views

What is $G_2(2^m)$, and how is it embedded in $\Gamma L_6(2^m)$?

I am trying to understand the classification of doubly transitive groups, specifically the nonsolvable affine case. Dixon and Mortimer (p.244) says there are three infinite families, one of which is $\...
3
votes
0answers
54 views

Torus in the small Ree group ${}^2G_2$ over an infinite field

In “Simple group of Lie type” by R. W. Carter there is a remark (after Theorem 13.7.4): It is not known whether $H^1$ coincides with the set of $\sigma$-invariant elements of $H$ if $\mathfrak{L}$ ...
4
votes
0answers
168 views

Are the centralizers of involutions in finite simple groups known?

By the famous result of Brauer and Fowler, there exist finitely many simple groups with given involution centralizer. There are many results which determine all finite simple groups with given ...