The finite-groups tag has no wiki summary.

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### Wielandt's exercise on regular groups [on hold]

Assume that the intransitive group $G$ has degree $n$ and minimal degree $n-1$.
If no transitive constituent of $G$ has degree $1$, then they all are faithful and all except one are regular.
Any ...

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272 views

### Geodesics in finite groups

It seems that I can generalize a result from compact, connected Lie groups to finite groups, but in order to do so, I need to have some kind of geodesics on finite groups.
Below is a proposition for ...

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185 views

### Equivalence classes of (2,3)-pairs in PSL(2,q)

This may not be a research-level question, which is why I submitted it to math.stackexchange first, but so far the question there has barely been viewed, let alone answered. Apologies if submitting it ...

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476 views

### Groups whose normal subgroups form a chain with respect to inclusion

Let G be a finite group. In general, given two normal subgroups N and K of G, we need not have N < K or K < N. The easiest example is the Klein 4-group V4 and its subgroups of order 2. So assume ...

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### $2$-group with two isomorphic normal subgroups of index $4$ with non-isomorphic quotients

Hypothesis: Let $P$ be a finite $2$-group with two isomorphic normal subgroups $M$ and $N$ such that $P/M\cong C_4$ (the cyclic group of order $4$) while $P/N\cong C_2^2$. By the lattice theorem, ...

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### Homomorphisms from irreducible spaces to reducible spaces

Let $P_{\lambda}$ be a Young symmetriser associated to the following tableau $(a_1 a_2 a_3 b_3 ; b_1 b_2)$ where the entries seperated by the ; belong to first and second COLUMNS of the tableau. Take ...

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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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195 views

### Minimum word length for an unusual set of generators of the symmetric group

Problem. Let $n\geq 2$ and let $T$ be the set of all permutations in $S_n$ of the form
$$t_k:=\prod_{1\leq i\leq k/2}(i,k-i) \qquad \hbox{for $k=3,4,\ldots,n+1$}.$$
Find the least integer ...

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### A question on p-groups, and order of its commutator subgroup [migrated]

$\textbf{QUESTION-}$
Let $P$ be a p-group with $|P:Z(P)|\leq p^n$. Show that $|P'| \leq p^{n(n-1)/2}$.
$\textbf{TRY- }$ If $P=Z(P)$ it is true. Now let $n > 1$, then
If I see $P$ as a nilpotent ...

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141 views

### Restricting the Steinberg representation of $SL_{2n}$ over a finite field to the symplectic group

Let $\text{St}_n(\mathbb{F}_q)$ be the Steinberg module (over $\mathbb{C}$) for $\text{SL}_n(\mathbb{F}_q)$.
What is the irreducible decomposition of the restriction of $\text{St}_{2n}(\mathbb{F}_q)$ ...

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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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618 views

### Are There Always Group Generators Which Give Unimodal Growth?

Suppose $G$ is a $k$-generated finite group. Is there always a set of $k$ elements which generate the group and have a unimodal counting function?
Background:
The counting function, $f(n)$, is a ...

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113 views

### The coproduct on the 2-boxes space of the goup-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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### Order-increasing bijection from arbitrary groups to cyclic groups

In his answer to this previous MO question, Gjergji Zaimi referred to the statement that for every finite group $G$ of order $n$, there is a bijection $\sigma \colon G \to \mathbb{Z}/n\mathbb{Z}$ ...

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### How many conjugacy classes of subgroups does GL(2,p) have?

How many conjugacy classes of subgroups does $\mathrm{GL}(2,p)$ have?
For instance the dihedral group of order $2n$ has $\tau(n)$ cyclic normal subgroups and $\sigma(n)$ "dihedral" subgroups (as ...

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### What's the ratio of inclusions of finite groups with a distributive lattice?

Definition: Two inclusions of finite groups are equivalent, $(A \subset B) \sim (C \subset D)$, if: $(A/A_B \subset B/A_B) \simeq (C/C_D \subset D/C_D)$ with $A_B$ the normal core of $A$ in $B$.
...

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132 views

### Is inner product preserved only by the stabiliser in a finite reflection group?

Is the following statement true for finite reflection groups?
Let $G$ be a finite reflection group acting on $\mathbb{R}^n$, let $x, y\in \mathbb{R}^n$
and let $z$ be in the orbit of $y$. If ...

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### An epimorphism into a profinite group

Let $p$ be an odd prime number, $G$ a finitely generated nonabelian profinite group, $L \lhd_o G$ a pro-$p$ group with $[G : L] = 2$. Suppose that there is a continuous surjection from $G$ onto a free ...

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### A question about minimal nonnilpotent groups

Let $G$ be a minimal nonnilpotent group with cyclic Sylow $p$-subgroup $P$ and normal Sylow $q$-subgroup $Q$ [see Huppert, Endlich Gruppen I]. If $Q$ is abelian and $q > 2$, then can we get that ...

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### Variety of nilpotent Lie algebras or $p$-groups

Here's a couple of analogous questions, one in terms of finite-dimensional complex Lie algebras and one in terms of finite $p$-groups; I'd be interested in an answer to either:
1) Let $\mathcal{L}$ ...

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### Dense free subgroups

Let $F$ be a free pro-$p$ group (for a prime number $p$) on a finite set $X$, $\Phi$ the abstract subgroup generated by $X$, $\{1\} \neq N \lhd_c F$. Is it possible that $\Phi \cap N = \{1\}$?

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### Lower central series in a free pro-p group

Let $F$ be a nonabelian finitely generated free pro-$p$ group, $H \leq_c F$ of infinite index. Denote by $\{F_n\}_{n \in \mathbb{N}}$ the lower central series of $F$, and set $r_n = [F : F_nH]$.
Is ...

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### A bound on the size of the center

Let $p$ be a prime number, $F$ a free pro-$p$ group, $H \leq_c F$ of infinite index. Can it be that $$\sup_{N \lhd_o F} |Z((F/N)/C_{F/N}(HN/N))| < \infty ?$$

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### Golod Shafarevich Inequality and Inequalities among higher Cohomology groups

As a consequence of Golod- Shafarevich, we get an inequality between second cohomology group of a $p$-group with coefficients in $F_p$ and the first cohomology group of a $p$-group with coefficients ...

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### Automorphisms of non-abelian groups of order p^3

There are two non-abelian groups of order p^3, namely, semi-direct product of Z/pZ x Z/pZ by Z/pZ and semi-direct product of Z/(p^2)Z by Z/pZ. What are the automorphism groups of these groups?

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### The cohomology group $H^{1}(GL_{2}(\mathbb{F}_{p}), M_{2}(\mathbb{F}_{p}))$

Let $M_{2}(\mathbb{F}_{p})$ be the vector space of 2$\times$2 matrices over the finite field $\mathbb{F}_{p}$ where $p$ is a prime number, and let $GL_{2}(\mathbb{F}_{p})$ be the group of invertible ...

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### The powers of non-empty subset of a group that generate a subgroup

If G is a group and A and B to non-empty subsets of G, then by AB we mean the set consist of all product ab where a is in A and b is in B.(Standard definition) Similarly we can define X^m where X is a ...

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### A normal form theorem for presentations of finite $p$-groups of nilpotency class $2$?

When constructing examples of nonabelian finite $p$-groups with abelian automorphism group (and certain other desired properties), the authors of papers like http://arxiv.org/pdf/1304.1974v1.pdf leave ...

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### (co)homology of symmetric groups

Let $S_n=\{\text{bijections }[n]\to[n]\}$ be the n-th symmetric group. Its (co)homology will be understood with trivial action. What are the $\mathbb{Z}$-modules $H_k(S_n;\mathbb{Z})$? Using GAP, we ...

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### Uniqueness of the direct product decomposition of inclusions of finite groups

This post is a generalization of Uniqueness of the direct product decomposition of finite groups.
Here we look inclusions of finite groups $(H \subset G)$ instead of just finite groups.
Definition: ...

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### In what sense is the classification of all finite groups “impossible”?

I think there is a general belief that the classification of all finite groups is "impossible". I would like to know if this claim can be made more precise in any way. For instance, if there is a ...

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### What are smallest finite images of triangle groups?

Given integers $a, b, c$ all at least 2, I would like to identify the smallest group with an element $x$ of order $a$, element $y$ of order $b$ such that the product $yx$ has order $c$. For example, ...

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### Classification of indecomposable inclusions $(H \subset G)$ with $G$ decomposable

Definition: A group $G$ is indecomposable if: $G = G_1 \times G_2 \Rightarrow \exists i \ G_i = 1$.
We can generalize the notion of indecomposable from groups to inclusion of groups as ...

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### Sums of degrees of irreducible complex characters

The sum of the degrees of the irreducible complex characters (not the square sum which is the group order) is relevant to determine the dimension of a maximal torus of the Lie algebra associated to ...

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### existence of a finite group which is the union of self normalizing subgroups

Can a finite group G be the union of self normalizing subgroups such that the intersection between any two of these subgroups is equal to the unit of the group G? I don't think so but I can't prove ...

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101 views

### Soluble group algebras and centralizers

Let $K$ be field with $\mathrm{char}(K) = p > 0$ and $G$ a finite group such that $KG$ is soluble. Then the $p$-Sylow-subgroup of $G$ is normal and contains the derived group of $G$ and every ...

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### breadth of a ﬁnite p-group

The breadth of an element $x$ in a ﬁnite $p$-group G is deﬁned to be that integer $b = br(x)$ such that $p ^b = |G : C_ G (x)|$, while the breadth $br(G)$ of $G$ is the
supremum of $\{br_ G (x) | x ...

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### $\mathcal{L}(H_i \subset G_i)$ distributive $\Rightarrow$ $\mathcal{L}(H_1 \times H_2 \subset G_1 \times G_2)$ modular?

Let $\mathcal{L}( G)$ be the lattice of subgroups of $G$ and $\mathcal{L}(H \subset G)$ the lattice of intermediate subgroups.
Definitions: A lattice $(L, \wedge, \vee)$ is distributive if, ...

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### Irreducible representation of Heisenberg group with characteristic 2?

As we all know that the irreducible representation for Heisenberg group can be classified easily when the group is over a finite field $\mathbb{F}_q$, where $q=p^n$ and $p$ is a prime greater than ...

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### Prime order elements in $GL(n,\mathbb{Z})$

What is known about the elements of prime order $p$ in $GL(n,\mathbb{Z})$? All I know at this point is that $GL(n,\mathbb{Z})$ has an element of prime order $p$ iff $p-1\leq n$ and that there are only ...

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### Existence of finite nonabelian groups satisfying certain identities

Is there a finite nonabelian group satisfying all of the following identities?
$$
(x^py^p)^2 = (y^px^p)^2, \quad p = 2,3,5,7,11,\ldots (\text{primes})
$$
I thank you all in advance.

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### Absolute irreducibility of a symmetric square?

This is a question I received today by email, which somebody more experienced with finite group representations can probably answer directly. Take $F:=\mathbb{F}_q$ for some prime power $q$, so ...

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### Questions on invariant operators of finite group representations

1) Is there an equivalent of the Casimir operator for an irreducible representation of a finite group?
2) Given an invariant operator of a certain group, can I check if it is invariant under only ...

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### Asymptotic density of finite abelian and solvable groups

For every natural number n, let:
Gn be the number of distinct group structures with at most n elements;
An be the number of distinct abelian group structures wit at most n elements;
Sn be the number ...

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### Special linear groups contained in symplectic groups

Let $q$ be a power of prime $p$, and $n, m, k$ positive integers such that $mk=2n$ and $2\leq m<2n$. Let $\mathrm{Sp}(2n,q)$ be the symplectic group of dimension $2n$ over $\mathrm{GF}(q)$ and ...

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### orthogonal group in characteristic 2

Let $O(2,\mathbb{Z}_2)$ be the orthogonal group of order two matrices. On $\mathbb{Z}_2$ there should exist just one odd quadratic form, hence the stabilizer subgroup $O^-$ of an odd quadratic for ...

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### Successive Schur covers

Let $G_0$ be a finite group and $G_j$ a Schur cover of $G_{j-1}$ for $j=1,2,3\ldots$. Is $G_2$ equal to $G_1$? If not, will the sequence stop after finite steps in general?

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### A question on conjugacy classes of central involutions in a finite group

An involution $a$ of a group $G$ is called central if there exists a sylow $2$-subgroup $H$ of $G$ such that $a \in C_G(H)$, or equivalently if the centralizer of $a$ contains a sylow $2$-subgroup.
...

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### Existence of homogeneous single chain compositions of a given maximal subfactor?

All the subfactors here are irreducible inclusion of hyperfinite II$_1$ factors.
A subfactor $(N \subset M)$ is Homogeneous Single Chain ($HSC$) if its lattice of intermediate subfactors is a ...

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### Combination of two recent problems about finite groups of square orders

Is there any finite group $G$ of order $n=m^2$ satisfying the following conditions?
(a) There is no subgroup of order $m$;
(b) There exist subsets $A$, $B$ such that $|A|=|B|=m$ and $G=AB$.
...