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**9**

votes

**2**answers

331 views

### Certain signed sum over $S_n$

The following question appeared in my research:
Let $G_1,G_2,G_3$ all be subgroups of $S_n$, and consider the sum
$$
\sum_{g_i \in G_i, g_1g_2g_3 = id} \epsilon(g_1)
$$
that is, we only consider ...

**3**

votes

**1**answer

153 views

### the number of indecomposable modules of finite groups over finite fields of a fixed dimension

I am interested in determining the the number of indecomposable modules of finite groups over finite fields of a fixed dimension. Specifically, I have the following conjecture:
Conjecture. Suppose we ...

**11**

votes

**0**answers

223 views

### Elements of order 3 normalizing no non-identity 2-subgroups in Almost Simple Groups

This question is partly motivated by a situation which arises in modular representation theory. A finite group $G$ is said to be almost simple if $G$ has a unique minimal normal subgroup which is a ...

**3**

votes

**0**answers

119 views

### Invariant Laurent polynomials under cyclic group action

Start with the cyclic group $G:=\mathbb{Z}/p$ of prime order $p$ and and an integer lattice $P:=\mathbb{Z}^p$. Let $G$ act on $P$ by cyclic permutation of coordinates. There is an induced action on ...

**2**

votes

**0**answers

120 views

### Special sets of involutions generating ${\rm S}_n$

For which positive integers $k$ and $r$ are there involutions $g_{n,i} \in {\rm S}_n$
$(n \in \mathbb{N}, \ i = 1, \dots, k)$ such that the following hold?:
for any $n$, the $g_{n,i}$ $(i = 1, ...

**7**

votes

**1**answer

112 views

### Covering a set with images of a transversal

Let $G$ be a permutation group on a finite set $\Omega$ with orbits $\Omega_1,\ldots,\Omega_k$.
By a transversal we mean a set $\lbrace\omega_1,\ldots,\omega_k\rbrace$ with $\omega_j\in\Omega_j$ for ...

**8**

votes

**1**answer

254 views

### Condition for a certain subset being a subgroup

For any finite group $G$ and $n$ a divisor of $|G|$, consider the following subset of elements of "co-order" dividing $n$:
$$G(n) = \{ g \in G \mid g^{|G|/n} = 1 \}$$
By a classical theorem of ...

**2**

votes

**1**answer

187 views

### $nse$ for which simple group was determined?

Let $G$ be a finite group and $\omega(G)$ be the set of element orders of $G$. Let $k\in\omega(G)$ and $m_k$ be the number of elements of order $k$ in $G$. Set $nse(G):= \{m_k : k \in\omega(G)\}$. ...

**2**

votes

**1**answer

113 views

### Conjugates and infinite index subgroups of free groups

Here I am asking for an analogue of Generating infinite index subgroups of a free group
Let $F$ be a nonabelian finitely generated free group, let $H \leq F$ be a finitely generated subgroup of ...

**0**

votes

**1**answer

273 views

### Number of involutions in a finite group [closed]

Let $G$ be a finite group. Does it possible to determine number of involutions in it? If not, is there any bound for it?

**6**

votes

**1**answer

297 views

### Are the distributive permutation groups linearly primitive?

An action of a group $G$ on a set $X \neq \emptyset$ is called transitive if $\forall x,y \in X$, $\exists g \in G$ such that $g.x = y$.
It is called primitive if it is transitive and preserves no ...

**1**

vote

**1**answer

192 views

### How can we conclude that $2p\nmid s_{2p}$?

Let $s_{2p}$ be the number of elements of order $2p$ in finite group $G$ and let $x$ be an element of order $2p$ in $G$. We can write $s_{2p}=\sum_{o(x)=2p}|x^G|$, where these conjugacy classes are ...

**1**

vote

**1**answer

147 views

### Does the hyperoctahedral group have only 3 maximal normal subgroups?

An hyperoctahedral group $G$ is the wreath product of $S_2$ and $S_n$, where $S_{n}$ is the symmetric group on $n$ letters, or in other words the semi-direct product $G=S_2^n\rtimes S_n$, w.r.t. the ...

**6**

votes

**1**answer

198 views

### What is the Schur multiplier of the affine linear group AGL(n,q)?

What is the Schur multiplier of the $n$-dimensional affine linear group $\mathrm{AGL}(n,q)$ over the Galois field with $q$ elements?
I am particularly interested in the simple case $n=1$. Computation ...

**8**

votes

**1**answer

280 views

### Characterization of Frobenius complements

I have learned that Frobenius complements are characterized (among finite groups) by having a fixed point free complex representation.
That is, a finite group $G$ is a Frobenius complement if and only ...

**0**

votes

**0**answers

24 views

### Is there a classification of finite nonabelian 2-groups with exponent 4? [duplicate]

Is there a classification of finite nonabelian 2-groups with
exponent 4?

**4**

votes

**1**answer

155 views

### A small rank linear combination of a small number of elements of a group

This is a version of
this question of Klim Efremenko.
Let $r>2$ be a natural number, say $r=3$ or $r=10$. Let $G$ be a finite group and $\rho$
be an irreducible complex representation of $G$. We ...

**0**

votes

**1**answer

167 views

### Two questions on the Schur multiplier of groups of order $p^4$

I tried to find a reference for the computation of the Schur multiplier of groups of order $p^4$.
The case in which $p=2$ is well known, see e.g. Table 1 at ...

**3**

votes

**1**answer

246 views

### Weyl groups of $E_6$ and $E_7$

The Weyl group $W_6$ of the Lie algebra $E_6$ is of order 51840, the automorphism group of the unique simple group of order 25920, while the Weyl group $W_7$ of the Lie algebra $E_7$ is of order ...

**2**

votes

**1**answer

90 views

### Generating subgroups of large index by a large chunk of a conjugacy class

Let $G$ be a finite simple group and let $C$ be a (non-trivial) conjugacy class of $G$. Let $H$ be a subgroup of $G$ such that $$|H\cap C| \geq \epsilon |C|.$$
Can one conclude that the index of $H$ ...

**3**

votes

**2**answers

188 views

### About the number of their conjugacy classes in some classes of finite simple groups

We know that the orders of simple groups $B_n(q)$ and $C_n(q)$ are equal. What about the number of their conjugacy classes? Are they equal or not?
Any reply, comment, remark or reference is ...

**-1**

votes

**1**answer

127 views

### Reference request: automorphism of abelian $p$-groups of rank 2

There is a result saying that if $P$ is an abelian $p$-group of rank $2$ then $$|Aut(P)|=(p-1)^kp^j(p+1)^r,$$
for some $k,j,r$ which depend on the order of $P$.
Moreover, if $P=C_{p^t}\times C_{p^s}$ ...

**2**

votes

**1**answer

130 views

### Measuring products of finitely generated subgroups of free groups

Let $F$ be a finitely generated free group, $H_1, \dots, H_n$ finitely generated subgroups of infinite index in $F$, and $\epsilon > 0$. Must there be an epimorphism to a finite group $\phi \colon ...

**8**

votes

**2**answers

300 views

### Can a positive measure subset of a free group be nowhere dense?

Let $F$ be a finitely generated free group and let $S \subseteq F$ be a subset for which there is some $\epsilon > 0$ such that for any epimorphism to a finite group $\phi \colon F \to G$ we have ...

**27**

votes

**2**answers

654 views

### Why do sporadic simple groups have so few conjugacy classes?

In finite group theory, there's a general intuition that the further away a group is from abelian, the fewer conjugacy classes it will have. So it is to be expected that non-abelian finite simple ...

**6**

votes

**0**answers

192 views

### Approximating Lie groups by finite groups

How can one approximate compact Lie groups by finite groups?
My wish is something like this:
Let $G$ be a compact Lie group.
There is a sequence of nested finite subgroups $G_n$ so that $G_n\to ...

**1**

vote

**0**answers

88 views

### Subgroups of the union of conjugates

This question is an attempt to find a version of this question with a positive answer. In this question, we ask if one can find big subgroups in the union of conjugtes of small subgroups of finite ...

**2**

votes

**1**answer

185 views

### Dihedral subgroups of $\mathrm{PSL}_2(\mathbb{F}_q)$

Let $\mathbb{F}_q$ be a finite field with $q=p^f$ elements. I need to know when $\mathrm{PSL}_2(\mathbb{F}_q)$ contains the group $D_{(q+1)/2}$, where by $D_n$ I mean the dihedral group of order $2n$. ...

**2**

votes

**1**answer

176 views

### Union of conjugates in p-groups

Fix a prime number $p$. Is there a sequence $\{a_k\}_{k \in \mathbb{N}}$ of real numbers with $$\lim_{k \to \infty} a_k = 0$$
such that for any finite $p$-group $G$, and any subgroup $H \leq G$ with ...

**1**

vote

**1**answer

118 views

### Decomposing quasi-finite separated group schemes

Let $U$ be a punctured disk, and let $G\to U$ be a quasi-finite separated group scheme. (Assume $K$ of char zero if it helps)
Why is $G = G_1\sqcup G_2$, where $G_1 \to U$ is finite and $G_2\to U$ ...

**4**

votes

**1**answer

135 views

### orders of maximal abelian subgroups

What are the orders of maximal abelian subgroups of the simple groups $F_4(q)$ and $C_4(q)$, where $F_4(q)$ is an exceptional group and $C_4(q)$ is a symplectic group?

**3**

votes

**1**answer

234 views

### Representation of GL(n, F_p) over F_p, for n small

The question is related to this post
Representation theory of the general linear group over a finite prime field
However, I am asking for more detailed references for n small, for example, for n=2, ...

**7**

votes

**2**answers

431 views

### What are the outer automorphisms of a Coxeter group?

I want to know the outer automorphisms of the Weyl group of $\mathrm{E}_8$, if any.
But I might as well ask the question more generally. Suppose we have a Coxeter diagram. This gives a Coxeter ...

**6**

votes

**1**answer

262 views

### Paper by I. N. Sanov, Solution of the Burnside problem for exponent 4

I have searched extensively online and for copies of printed journals containing the paper which details Sanov's solution to the Burnside Problem for exponent 4, which is widely cited in many papers ...

**3**

votes

**2**answers

221 views

### Smallest non-trivial conjugacy classes in simple groups and classes of involutions

I am interested in finding the size of the smallest non-trivial conjugacy class
of the simple groups $PSL(d,q)$ with $d>2$, $Sz(q)$ with $q>2$ and $R(q)$ with $q>3$.
My first question is ...

**7**

votes

**2**answers

238 views

### Is the free abstract group residually of rank d > 2?

Let $d \geq 2$ be an integer, and let $\mathcal{F}_d$ be the family of finite groups such that $G \in \mathcal{F}_d$ if and only if every subgroup of $G$ can be generated by at most $d$ elements.
Is ...

**5**

votes

**0**answers

208 views

### In general, are 'Young symmetrisers' given by Littlewood-Richardson 'Orthogonal projection Operators'?

Consider $V^{\otimes n}$ where $V$ is vector space and the representation of GL(V) acting in the usual way. Now if I consider tensor products or plethysms of irreducible spaces, this is not in general ...

**4**

votes

**0**answers

148 views

### Is there a nontrivial profinite word which is trivial in any group with at most d generators?

Let $F$ be a free profinite group of rank $\aleph_0$, and let $d \in \mathbb{N}$.
Let $N_d \lhd_c F$ be the intersection of all open normal subgroups $L \lhd_o F$ for which $F/L$ can be generated by ...

**4**

votes

**1**answer

397 views

### Generalization of a property of $A_n; n\geq 5$

Let $H$ and $K$ be two proper non-trivial subgroups of the
alternating group $A_n; n\geq 5$.
Then there exists a maximal subgroup $M$ of $A_n$
such that $H\not\leq M$ and $K\not\leq M$.
To see this
...

**3**

votes

**1**answer

156 views

### On the size of centralizers in a non-abelian finite simple group

It is known that for a finite non-abelian simple group $G$ we have $|G|<|C_G(x)|^3$ for some involution $x$. Is there a better bound for the order of centralizer of a nontrivial element of $G$ (not ...

**1**

vote

**1**answer

165 views

### Can we say that $A$ is a complement for a group $G$?

Let $A$ be a frobenius complement for a group $G$ i.e. $A$ act on $G$ by automorphism s.t. $C_A(g)=e$ for all nonidentity $g$.
Now, Action of $A$ can be linearly extended so that $A$ act on $F[G]$. ...

**2**

votes

**0**answers

128 views

### Irreducible representations of $Sp(4,\mathbb{F}_2)$

I'm trying to construct the irreducible representations (over $\mathbb{C}$) of the finite group $Sp(4,\mathbb{F}_2)$.
Using GAP, the character table is as follows:
$$
\left(\begin{matrix}
1 & 1 ...

**1**

vote

**0**answers

174 views

### Rational conjugation of elements of a finite group

Let $G$ be a finite group. Two elements $x$ and $y$ of $G$ are said to be rationally conjugate, written $x \sim_{r} y$, if and only if $\langle x\rangle$ and $\langle y\rangle$ are conjugate subgroups ...

**4**

votes

**1**answer

162 views

### Decomposing representations of finite groups of Lie type via computer

This is related to my previous question here.
Let me remind you what that question asked:
Let $\text{St}_n(\mathbb{F}_q)$ be the Steinberg module (over $\mathbb{C}$) for ...

**5**

votes

**3**answers

184 views

### bounding from below the cardinality of a set of generators of the $n$-fold cartesian product group of a finite group

Let $G$ be a non-trivial finite group. Let $n\in\mathbf{Z}_{\geq 1}$ and let $G^n$ be the $n$-fold cartesian group product of $G$. Let $S\subseteq G^n$ be a generating set of $G^n$.
Q: Is $|S|\geq ...

**9**

votes

**2**answers

245 views

### Vertex-primitive graphs with two vertices having almost the same neighbourhood

Hypothesis: Let $\Gamma$ be a vertex-primitive graph with two vertices $u$ and $v$ such that $$|N(u) \cap N(v)|=|N(v)|-1$$
Question: Is it true that $\Gamma$ must either be a complete graph or have ...

**1**

vote

**0**answers

122 views

### Explicitly showing that a free group is LERF [closed]

Let $F$ be a free group on a finite set $X$, and let $M$ be a finitely generated subgroup.
Marshall Hall's theorem states that $M$ is closed in the profinite topology on $F$. That is, $M$ is the ...

**1**

vote

**2**answers

235 views

### Rational Conjugacy Classes of Finite Groups

Suppose $G$ is a finite group and $A$ is the set of all character values of $G$. By character values, I mean entries of the character table of $G$. Let $\Gamma = ...

**6**

votes

**3**answers

325 views

### Restriction of scalars for the adjoint representation of $SL_2(\mathbb F_q)$

Let $p$ be a prime number, $q=p^e$ a power of $p$, and $G=SL_2(\mathbb F_q)$. Let $V$ be the adjoint representation of $G$, i.e. $V$ is the 3-dimensional $\mathbb F_q$-space of of (2,2)-matrices of ...

**13**

votes

**1**answer

428 views

### Why is Klein's representation of $PSL_2(\mathbb{F}_7)$ hard to obtain?

In his famous article [1] Klein constructs a representation of $G=PSL_2(\mathbb{F}_7)$ in $\mathbb{C}^3$ (of which the first invariant polynomial of three variables gives rise to the famous Klein's ...