The finite-groups tag has no wiki summary.

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### Fibres of square map on finite groups and the inverse set of a subset

Let $G$ be a finite group and $q:G\rightarrow G:g\mapsto g^2$ the square map. Now, if $A$ is a subset of one fibre of $q$, i. e.
$$a^2=b^2$$
holds for all $a,b\in A$, is there always some $g\in G$ ...

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### When is $S_n \times S_m$ a subgroup of $S_p$?

I asked the following question on math.stackexchange several months ago:
Let $n,m,p>1$ be such that $S_n \times S_m \hookrightarrow S_p$. Does it imply that $p \geq n+m$?
Derek Holt gave a ...

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### Paths in groups

Given a finite group $G$, write $K(G)$ for the complete digraph on the elements of $G$. Label the edge from $g$ to $h$ by element $g^{-1}h$.
Question: For what groups does there exist a Hamiltonian ...

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### on the extensions of $ A_5$ by $A_5$ [closed]

Let $G$ be a finite group such that $G$ has a normal subgroup $H$ and $H$ is isomorphic to the alternating group $A_5$. Also we know that $G/H \cong A_5$.
Can we say that $G \cong A_5\times A_5$?
...

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### Fast construction of straight line programs?

Given a group $G$ and a set of generators $A$, we can ask ourselves (and do ask ourselves all the time) to bound the diameter of $G$ with respect to $A$. The diameter, let us recall, is defined to be ...

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### Aschbacher's description of Sylow 2-subgroups

I'm looking for a few more examples of using Aschbacher (1980)'s “fundamental SL2 subgroups” description of Sylow 2-subgroups of finite groups of Lie type in odd characteristic. I do not yet ...

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### A characterization of almost relatively free, finite $p$-groups

Let $G$ be a finite minimally $d$-generated $p$-group.
If $G$ is relatively free, that is $G$ is a quotient of the free group $F$ on $d$ generators by a fully invariant subgroup of $F$, then the ...

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### A question on $p$-central $p$-groups

Let $p$ be a fixed prime. A group $G$ is termed $p$-central if every element of order $p$ in $G$ lies in the center.
Having a finite $p$-group $G$ of rank $k$ (the least integer, such that every ...

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### On the groups of order $p(p^2+1)$

Let $G$ be a group of order $p(p^2+1)$, where $p$ is an odd prime number and $p>3$. Easily we can see that $G$ is solvable and so $G$ has a Hall subgroup $L$ of order $p^2+1$. Also we know that ...

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### “A locally dual polar space for the Monster”

I am currently looking at Ronan and Stroth's 1984 paper Minimal Parabolic Geometries for the Sporadic Groups. When considering the $3$-minimal parabolic system of $F_{1}$, they cite a preprint by ...

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### Questions about some finite p-groups of coclass 2

I have two questions about a specific type of finite $p$-groups that i've seen in an interesting work on automorphisms of finite p-groups.
Let $G$ a finite $p$-group of order $p^{n}$ such that:
- ...

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### Projective characters with corresponding factor set

The following is just a follow up to my previous question. I have a finite group $H$ with 14 ordinary characters. The Schur multiplier $M(H)\cong 2^2$. Hence the group $H$ will have 3 sets of ...

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### Generating random finite groups

I would like a method to efficiently generate a random finite group of a given order $n$.
If there are $g(n)$ non-isomorphic groups of order $n$,
ideally each group would occur with probability ...

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### How do I verify the Coq proof of Feit-Thompson?

I probably don't have the appropriate background to even ask this question. I know next to nothing about formal or computer-aided proof, and very little even about group theory. And this question is ...

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### Sylow 3-subgroups of symmetric group

Is there any routine technique to find a set of permutations which generate a Sylow $3$-subgroup of the symmetric group $S_{n}$?

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### Decomposing representations of finite groups

Let $G$ be a finite group, $p$ a prime number. We denote by $\mathbb{F}_p$ the field of cardinality $p$. Let $V$ be an infinite dimensional representation of $G$ over $\mathbb{F}_p$.
Must there be ...

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### Is there a precise notion of “almost all” such that almost all finite groups are Galois groups of extensions of the rationals?

I vainly tried to define a notion of "almost solvable group" such that every "almost solvable group" is the Galois group of a finite extension of the rationals, but I can't figure out the right way to ...

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### Maximum length of chains of subgroup in GL(n,q)

Let G=GL(n,q) be a general linear group n-dimensional over a field with q element (q power of a prime). I am looking for an estimate of maximum length of chains of subgroup in G.
Thanks.

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### Coprime automorphisms of finitely generated pro-$p$ groups

Let $P$ be a finitely generated pro-$p$ group and let $G$ be a semidirect product $P \rtimes A$, where $A$ is a finite group of order coprime to $p$ that acts faithfully on $P$. Then one can show ...

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### Groups like symmetric group

For sufficiently large n consider this question
Let $G$ be a finite group with the following properties:
$|G|=n!$
$H,K$ are subgroups of $G$ such that $H\cap K=1$ and $H\cong S_{3}$ and $K\cong ...

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### Automorphisms of profinite groups

Let $d,n \in \mathbb{N}$, and $p$ a prime number. Let $F$ be a free pro-$p$ group on $d$ generators. Is there an automorphism of $F$ of order $n$?

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### generality of the lattice of normal subgroups

Let $(X,\le)$ a (finite) modular lattice. Is there a (finite) group $G$ such that the lattice of all normal subgroups of $G$ is isomorphic to $(X,\le)$?

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### Which finite groups can be characterized by their subgroup orders?

Given a finite group $G$, we denote by $\pi_s(G)$ the set of orders of its subgroups. Which finite groups $G$ can be characterized by the set $\pi_s(G)$, i.e. $\pi_s(H)=\pi_s(G)$ implies $H\cong G$? ...

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### Subgroups from which all class functions extend to class functions on the ambient group

Until someone suggests better terminology, let me call a subgroup H of a finite group G segregated if every class function on H can be extended to a class function on G. Equivalently, H should have ...

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### Properties to have matrices that commute in $\mathrm{GL}_n(\mathbb C)$

Let $G$ be a finite subgroup of $\mathrm{GL}_n(\mathbb C)$, $A,B \in G$ whose eigenvalues are thus in the unit circle.
Assume that the eigenvalues of $A$ are included in a circle arc of ...

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### What algebras does the hidden subgroup problem for finite abelian groups apply to?

Shor's algorithm is said to solve the hidden subgroup problem for finite Abelian groups, of which factoring and the discrete logarithm problem for integers belong to. Apparently the HSP for finite ...

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### Bound for the Frattini subgroup of a $p$-group

Assume that $G$ is a finite $p$-group, $p$ odd, with a non-trivial elementary abelian Frattini subgroup. Then both $\Phi(G)$ and $G/ \Phi(G)$ are vector spaces over $\mathbb{F}_p$. Is it possible to ...

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### Normal Subgroup Growth

Let $F$ be a free group on $d$ generators.
Denote by $F_{k}$ the $k$-th term in $F$'s derived series. Put $G = F/F_k$. What is the normal subgroup growth of $G$?
Explicitly, for each natural number ...

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### Simple groups and words

Let S be a finite simple nonabelian group, w a word in a finite number of variables which is not a power of another word. Must there be a substitution of elements of S in w such that the resulting ...

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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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### Non split extension isomorphic (as a group) to a split extension

$\def\Z{\mathbb{Z}}$
Let $A$ be a finite abelian group and $G$ a finite group acting on $A$.
Then the extension $0 \to A \to E \to G \to 1$ is splits if and only if the corresponding $2$-cocycle is ...

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### Connection between two theorems on character values?

In a recent arXiv preprint here, Dipendra Prasad has revisited a 1976 theorem of Kostant (Theorem 2 in the paper On Macdonald's $\eta$-function formula, the Laplacian and generalized exponents, ...

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### Two $p$-groups whose automorphism groups have isomorphic Sylow $p$-subgroups

Fix a prime $p$, and let $M$ be the unique nonabelian group of order $p^3$ and exponent $p$. Let us denote by $E_n$ the elementary abelian group of rank $n$.
Is it true that $\operatorname{Aut}(M ...

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### Action of automorphism group on Lie algebra [closed]

I want to know whether an automorphism group of a simple Lie algebra over $GF(2)$, acts transitively on non-zero elements of Lie algebra or not? How can I check this property?

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### Control of $p$-extensions by subgroups of index coprime to $p$

Let $G$ be a finite group and let $M$ be a $G$-module that is a finite abelian $p$-group. Suppose we have extensions
$1 \rightarrow M \rightarrow E_1 \rightarrow G \rightarrow 1$
and
$1 ...

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### a problem in character theory of finite group

I want to ask a question on character theory of finite group.
Let $G$ be a finite nonablian group， and let $p$ be the minimal divisior of |G|=n. Suppose $\chi\in IRR(G)$. Let $A=Gal(Q_n/Q)$. $A$ ...

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### Orbits of stabilizer of two points in a 2-transitive permutation group

I was doing something which needs to know sizes of all orbits of the stabilizer of two points in a 2-transitive permutation group. Since all 2-transitive permutation groups are known ans so are their ...

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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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### In which fixed-point free representations is the sum of every 3 elements invertible?

A representation $\rho:G\to GL_k(\mathbb{F})$ is called fixed-point free if for every $1\neq g\in G$ and every $0\neq v\in \mathbb{F}^k$, $\rho(g)v\neq v$. Stated differently, it is a representation ...

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### Primitive, non-2-transitive groups with very large orbitals?

Let $G$ be a transitive permutation group on a set $X$ with $n$ elements. Assume that $G$ is primitive, i.e., $G$ preserves no non-trivial partition of $X$. Assume as well that $G$ is not ...

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### On the Complement of a subgroup

This question was asked in
http://math.stackexchange.com/questions/729648. Since I did not get any answer I am asking it here.
In an answer in Mathoverflow I see an answer but I could not ...

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### Real representation of group of odd order

Let $G$ be a finite group of odd order. Suppose that $G$ has a real 4-dimensional faithful representation. Is it true that $G$ should be abelian in this case?

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### Finite groups $G$ so that $G$ has exactly two subgroups of a given order

Is there a finite group $G$ and a divisor $d$ of $|G|$ so that $G$ contains exactly two subgroups of order $d$?
The motivation for this question is an old qual problem (see ...

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### Are the following Cayley digraphs Hamiltonian?

Consider the Cayley graphs $A'G_n$ on the alternating group $A_n$ with generating set $S = \{(1i2) : 3 \leq i \leq n\}$, for $n \geq 4$.
See the following page on Alternating Group Graphs for ...

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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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### Double coset relation for unique intermediate subgroup (with homogeneity)

Let $G$ be a group and $H$ a subgroup such that there is a unique (non-trivial) intermediate subgroup $K$ (i.e. $H < S < G$ implies $S=K$) and $(H \subset K) \sim (K \subset G)$ (homogeneity) ...

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### On a metric associated to certain finite groups

Let $G$ be a finite group. Define $d:G\times G\longrightarrow\mathbb{N}$ by $d(x,y)=o(xy^{-1})-1, \forall\, x,y\in G$. Then $d$ is a metric on $G$ if and only if
$$(*)\hspace{5mm}o(ab)<o(a)+o(b), ...

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### Max order for which connected Cayley Graphs are known to be Hamiltonian

There is a well-known conjecture that all connected Cayley graphs are Hamiltonian.
For how large a value of n has the conjecture been verified (i.e., for all groups whose order is at most n)?

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### Are the homogeneous single chain subfactors, Dedekind?

Background: See here and there.
Recall that a subfactor is Dedekind if all its intermediate subfactors are normal.
A subfactor $(N \subset M)$ is Homogeneous Single Chain (HSC) if its lattice ...

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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 $G$.
From $G/Z(G)\cong Inn(G)$ we know complete group is the anewer for the simplest case, though this class ...