The finite-groups tag has no wiki summary.

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### p-groups and 2-generated abelian images

Let $p$ be a prime number. Is there a finite nonabelian $p$-group $G$ such that any finite epimorphic $2$-generated image of $G$ is abelian?

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### Dualization of a theorem of Øystein Ore

This post is a dualization of Generalization of a theorem of Øystein Ore in which we have proved:
Theorem: Let $(H \subset G)$ be an inclusion of finite groups such that the lattice ...

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### Generalization of a theorem of Øystein Ore in group theory

Theorem (Øystein Ore, 1938): A finite group $G$ is cyclic iff its lattice of subgroups $\mathcal{L}(G)$ is distributive.
Proof: see below.
Let $(H \subset G)$ be an inclusion of finite groups and ...

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### Asymmetry of functions defined on the $n$-th roots of the unity

Let $\mathcal{A} = \{V : \mathbb{U}_n \rightarrow \mathbb{C}\}$ where $\mathbb{U}_n$ is the group of the complex $n$-th roots of the unity. This group naturally acts on $\mathcal{A}$: for any $a \in ...

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### Algorithm to check is representation irreducible ? Algorithm to decompose the reducible one ?

Question 1 Given a representation of a finite group, what algorithm can be used to check is it irreducible or not ?
(Main case - complex numbers, comments on other cases are also welcome. "Given" ...

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### 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 ...

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### Schur covering group for S4

It is well-known that the symmetric group S4 has two Schur covering groups, S4-tilde and S4-hat. There are explicit presentations for both groups, and we know that S4-hat is isomorphic to GL(2,3). ...

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### 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 ...

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### 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 ...

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### 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 ...

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### Finite quasigroup not coming from a scrambled finite group

Given any finite group with multiplication $m(-,-)$ and three permutations $p,q,r$ on the underlying set of the group, we can obtain a quasigroup with binary operation $g*h:= p(m(q(g),r(h)))$. What is ...

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### 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 ...

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### 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 ...

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### Mclaughlin Graph

how can i construct a strongly regular graph with parameter $(275,112,30,56)$(Mclaughlin Graph), (105,32,4,12)?
I need adjacency matrix of them?
I know they are unique.

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### 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, ...

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### 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 ...

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### 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 ...

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### Is this statement true? [closed]

Let $G$ be a finite group such that $p\mid |G|$ and $p^2\nmid |G|$, where $p\geq3$ is a prime number. Is it true that $G$ is a direct product of simple groups? why?

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### Uniform proof that a finite (irreducible real) reflection group is determined by its degrees?

Given a finite (irreducible real) group $G$ generated by reflections acting on euclidean $n$-space, it was shown by Chevalley in the 1950s that the algebra of invariants of $G$ in the associated ...

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### $nse$ 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 $nse(G):= \{m_k : k \in\omega(G)\}$. ...

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### 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 ...

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### 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?

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### 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 ...

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### 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 ...

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### 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 ...

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### Special automorphisms of extraspecial groups

Let $G$ be an extraspecial group of order $p^{2r+1}$ (where $p$ is an odd prime), and let $V$ be a faithful representation of $G$. Consider the normal subgroup $H$ of $Aut(G)$ consisting of all ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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?

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### 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 ...

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### The coproduct on the 2-boxes space of the group-subgroup subfactor planar algebras

Let $(H \subset G)$ be an inclusion of finite groups.
Let the subfactor $(\mathcal{R} \rtimes H \subset \mathcal{R} \rtimes G)$ with $\mathcal{R}$ the hyperfinite ${\rm II}_1$ factor, and its planar ...

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### 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 ...

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### 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 $P=C_{p^t}\times C_{p^s}$ ...

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### 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 $H$ ...

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### Is a group uniquely determined by the sets $\{ab,ba\}$ for each pair of elements a and b?

This is a cross-posted question, originally active here on math.stackexchange.
For a given group $G=(S,\cdot)$ with underlying set $S$, consider the function
$$
F_G:S\times ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 $2n$. ...

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### 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 ...

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### Find finite groups $G\cong Aut(G)$

I become interested in this problem because $G\cong Aut(G)$ suggests a special symmetry in $G$ (This kind of group describes its own symmetry).
From $G/Z(G)\cong Inn(G)$ we know complete group is the ...

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### 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 $G_2\to U$ ...

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### 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 ...

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### 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 ...

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### 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?

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### 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, ...

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### 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 ...