# Questions tagged [finite-groups]

Questions on group theory which concern finite groups.

**10**

votes

**2**answers

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

**0**answers

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

**1**answer

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

**3**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**4**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**3**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**2**answers

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

**1**answer

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

**0**answers

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

**2**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**2**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**2**answers

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

**3**answers

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

**1**answer

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

**1**answer

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

**3**answers

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

**2**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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 ...