# Questions tagged [finite-groups]

Questions on group theory which concern finite groups.

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### Cyclic subgroups of finite $p$-groups

Let $G$ be a finite non-Dedekind $p$-group with non-cyclic center, where $p$ is an odd prime.
By $[\langle x\rangle]_G=\{g^{-1}\langle x\rangle g\ |\ g\in G\},$
I mean the conjugacy class of the ...

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### A lower bound on the number of fixed points of an inner diagonal automorphism of ${^2}{\operatorname{E}_6(q^2)}$

Let $q=p^f$ be a power of the prime $p$ and let $\alpha$ be an inner diagonal automorphism of ${^2}{\operatorname{E}_6(q^2)}$, i.e., an element of $({^2}\operatorname{E}_6)_{\mathrm{ad}}(q^2)$ in the ...

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### $p$-groups in which all normal abelian subgroups are cyclic

It is well-known that any finite $p$-group in which all its abelian subgroups are cyclic is either a cyclic group or a generalized quaternion group.
What can be said about $p$-groups in which every ...

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### Finite subgroups of classical groups over $\mathbb C$

I'm interested in examples of "big" finite subgroups of $G(\mathbb C)$ for $G=\mathrm{Sp}_{2n}, \mathrm{SO}_{2n+1}$. A subgroup $H$ of $G(\mathbb C)$ is said to be big if the associated representation ...

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### Deduce Schur-Zassenhaus theorem from Wedderburn-Malcev theorem

Is it possible to derive Schur-Zassenhaus theorem from Wedderburn-Malcev theorem? For the existance part I have the following idea: Let $G$ be a finite Group and $N$ an abelian Hall $p$-subgroup. Let $...

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### Largest almost quasisimple group that acts on a spin module

I'd like to know the largest almost quasisimple group $G$ which acts faithfully and irreducibly on the spin module for an odd dimensional orthogonal group $\Omega(2k+1,q)$, or on each of the two half-...

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### How do I know if an irreducible representation is a permutation representation?

I have a vague question, a less vague question and a lot of vaguer questions about permutation representations of a finite group $G$.
Vague question. Recall that if $G$ acts on a finite set $X$, we ...

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### Counting elements having a given cycle structure in maximal subgroups of a generalized symmetric group

Let $G$ be the wreath product $C_7\wr S_{18}$, where $C_7$ is the cyclic group of order 7 and $S_{18}$ is the symmetric group on 18 symbols. Consider $G$ to be embedded in the group $S_{126}=S_{7\cdot ...

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### recognition of symmetric groups in GAP

In GAP (https://www.gap-system.org), there is a function IsSymmetricGroup, which tells you whether a subgroup of $S_n$ generated by given permutations is all of the $S_n$. It looks like it takes ...

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### Generating symmetric groups with small cycles

This was asked but never answered at MSE.
Let $S_n$ denote the symmetric group and let $H$ be a subgroup which contains
an $n$-cycle. If $n$ is prime, and if $H$ also contains a 2-cycle, then ...

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### About the eigenvectors of a matrix related to a Cayley graph

In some papers about the cayley graphs of finite groups the behaviour of the eigenvalues and eigenvectors of $\phi$ were discussed when $\phi=\sum_{g\in G} \lambda_G(g)$ and $\lambda_G(g)$ is defined ...

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### More on finite groups generated by involutions

Can finite groups, which are generated by involutions, be represented as a quotient of a Coxeter group?

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### Prove some inertia group $T$ is a $p'$-group

Let $G$ be a finite group and $p\in\pi(G)$. Suppose that
$\quad$(i) for any non-principal $\chi\in\mathrm{Irr}(G)$, $p\nmid\frac{|G|}{|\mathrm{ker}\chi|\chi(1)}$;
$\quad$(ii) $E$ is the unique ...

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### Are finite nilpotent groups the only finite groups with abelian Frattini quotient?

It is obvious that if the Frattini quotient of a finite group $G$ is abelian, then $G$ is abelian by nilpotent and that finite nilpotent groups have abelian Frattini quotient.
I wonder if there is ...

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187 views

### Portability of Thompson theorem about solvability to Moufang loops

Say we have a finite Moufang Loop $Q$, $|Q|<\infty$.
There is a theorem proved by Thompson that states:
Group $G$, $|G|<\infty$ is solvable $\iff$ $\forall a, b \in G \langle a, b\rangle$ is ...

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### About the degree of character of $PSL(n,q)$

It is well known that for $n\geq2$ the group $PSL(n,q)$ is simple except for $PSL(2,2)=S_3$ and $PSL(2,3)=A_4$.
Let $G$ be one of the simple groups $PSL(n,q)$. From the ATLAS of finite group, we ...

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### Chains of right annihilators in group rings

See the update below
This problem emanates from a question on not-so-simple random walks on finitely generated groups. But to explain the connections would require an extremely long essay.
Let $G$ ...

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### Can we classify all finite 2-generated groups $G$ such that if $x,y$ generate $G$, then so does $x,yxy^{-1}$?

Can we classify finite 2-generated groups $G$ satisfying the following property:
For any pair $x,y$ which generate $G$, the pair $x,yxy^{-1}$ also generates $G$.
By the comments, no nontrivial ...

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### Testing whether two elements of $\text{SL}(2, \mathbb{F}_{2^n})$ generate the entire group

Given two elements $A,B \in \text{SL}(2, \mathbb{F}_{2^n})$, is there a (computationally inexpensive) test one could perform to check whether together they generate the entire group?

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### How many symmetric strings a permutation fixes

Let $A$ be an alphabet of $N$ symbols. Let $S_n$ be the group of permutations of $n$ symbols. A permutation acts on a string of letters from $A$ in the obvious way.
If I ask, given a permutation $\pi\...

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### On the Upper Density of $C_2$ in finite groups

We define the upper density $\rho (G)$ of a finite group $G$ as the ratio of the number of finite groups of order $<n$ which contain a subgroup isomorphic to $G$ to the number of groups of order $&...

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### Upper Bound Lemma implies the Ergodic Theorem for Random Walks on Groups?

Cross-Posted from Math Stackexchange.
Ergodic Theorem A random walk on a finite group $G$ driven by a probability $\nu\in M_p(G)$ is ergodic if $\operatorname{supp}(\nu)$ is not concentrated
on a ...

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### Module with indecomposable and decomposable reductions mod $p$

Let $G$ be a finite group and $p$ a prime dividing the order of $G$. Let $V$ be an irreducible $\mathbb{C}[G]$-module.
Let $F$ be the finite field $\mathbb{Z} / p \mathbb{Z}$. Suppose that there ...

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### Conjugacy of powers of elements in $PSL_n(\mathbb{F}_{\ell})$

Is the following true for some prime $p$?
There exists some prime $\ell$ and some $n$ such that $PSL_n(\mathbb{F}_{\ell})$ contains nontrivial $p$-torsion, and moreover if $x \in PSL_n(\mathbb{F}_{\...

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### Words Growth in Finite Groups

Let $G$ be a finite group with a set of generators and let $\Gamma$ be its Cayley Graph. Let $b_k$ be the number of elements in the ball of radius $k$. I am interested in what is known about the ...

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### Quillen + construction for finite groups

Is there an example of two non isomorphic finite groups $G$ and $H$ such that $BG^{+}$ is homotopy equivalent to $BH^{+}$ ?

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### An example of a finite group with some specific permutable subgroups

The following question is about finite groups.
Let $G$ be a finite group and let $H, K \leqslant G$. We say that $H$ permutes with $K$ if $HK = KH$ and in this case $HK \leqslant G$.
The Symbol $\pi ...

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### Symmetries of irregular simplices

On the wikipedia page of tetrahedron, there is a list of eight symmetry groups for a (possibly irregular) $3$-simplex (with unmarked faces). There is also a list on the page of 5-cell but doesn't ...

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### Is group cohomology with the inversion action order two?

Let $\mathbb{Z}^\text{inv}$ denote the $\mathbb{Z}/2$-module defined by the inversion action on $\mathbb{Z}$. Let $G_0$ be a finite group that acts trivially on $\mathbb{Z}^\text{inv}$. Then one can ...

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### Classification of finite HNN-extensions of a finite group with respect to an isomorphism between cyclic subgroups

Given the data of a triple $(G,h,k)$ where $G$ is a finite group, and $h,k\in G$ of the same order which together generate $G$, I'm interested in understanding the possible pairs $(i,\alpha)$, where $...

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### Motivation for the definition “strongly real element” in group?

Elements in finite groups can splitted in real / non-real elements.
Which is quite well-motivated definition:
element is called real if all characters take real values on it.
Equivalent requirment ...

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### Bounding the exponent of finite $p$-groups with normalizer conditions on cyclic subgroups

Suppose $P$ is a non-cyclic finite $p$-group satisfying the following two conditions:
All cyclic subgroups of order $p$ in $P$ are normal (this is equivalent to saying that $\Omega(P) \subset Z(P)$).
...

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### Translating parabolic induction as $\Lambda G^F/U^F\otimes_{\Lambda L^F}-$ to $\hom_{\Lambda L^F}(\Lambda U^F/G^F,-)$?

Suppose $P=L\ltimes U$ is an $F$-stable parabolic subgroup of a finite group of Lie type $G$, with $F$-stable Levi complement $L$. Here $F$ is a Frobenius endomorphism, and $G^F$ is the subgroup of ...

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### Generating sets of the symmetric group that yield isomorphic Cayley graphs

Let $S$ and $S'$ be subsets of size $k$ of $\mathfrak{S}_n$.
Are there any necessary or sufficient conditions to determine whether or not $S$ and $S'$ yield isomorphic Cayley graphs?
Assuming we ...

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### Automorphism group of a graph

Suppose $\Gamma$ is a simple graph and $G=\mathrm{Aut}(\Gamma)$ is the automorphism group of $\Gamma$. If $G$ stabilizes a subgraph $\Gamma_1$,, and $G_0$ is the point-wise stabiliser of the set $V(\...

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### Non-Boolean Eulerian interval of finite groups

An Eulerian subgroup lattice is Boolean (see here), so it is natural to wonder whether it is also true for an interval of finite groups. The smallest non-Boolean Eulerian lattice is the following:
It ...

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### Is an Eulerian subgroup lattice boolean?

Let $G$ be a finite group and $\mu$ the Möbius function of the subgroup lattice $\mathcal{L}(G)$.
The reduced Euler characteristic of the order complex of the coset poset $\{ Kg \ | \ K<G, \ g \...

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### What classifies involutive automorphisms on finite groups? What classifies involutions on finite based rings?

Groups
Let $G$ be a finite group. An involutive automorphism on $G$ is an automorphism $i\colon G \to G$ such that $i^2 = 1_G$.
Question 1. What classifies involutive automorphisms on a given (non-...

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### Automorphism group of $UT(n,p)$, the group of unitriangular matrices over the field $\mathbb{F}_p$

I am interested in understanding (at least roughly, if no such a description exists) the group of automorphisms for the group $UT(n,p)$, of unitriangular matrices over the field $\mathbb{F}_p$ on $p$-...

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### Automorphism group of a special commuting graph

Suppose $S_6$ is the symmetric group on six letters and let $X$ denote the conjugacy class containing $(12)(34)$. Define a graph $\Gamma$ with vertex set $X$ and edges precisely the 2-element subsets ...

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### Construction of representations of the Mathieu groups?

The Mathieu groups are beautiful simple finite groups. (They were the first sporadic groups to be discovered in 1861-1870).
They are related with many other miraculous constructions in mathematics:
...

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### Fullrankness of sum of time shifts

I am working with finite Gabor frames and in this context a problem appeared which I am trying to solve for a couple of weeks now.
Given a $(p,k,1)$ cyclic difference set for $\mathbb{Z}_p$ which is ...

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### A stronger version of a problem of Kenneth Brown using representations

Let $G$ be a finite group and $\mathcal{L}(G)$ its subgroup lattice. Let $\mu$ be the Möbius function on $\mathcal{L}(G)$.
The reduced Euler characteristic of the order complex of the coset poset $\{ ...

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### When is the rank 2 free metabelian group of exponent $n$ center free?

Let $M_n$ be the rank 2 free metabelian group of exponent $n$. For which $n$ is $M_n$ center-free?
The abelianization $M_n^{ab}\cong C_n\times C_n$, so the commutator subgroup $M_n'$ is a cyclic $(\...

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### Analogy between product of conjugacy classes and irreps: is there analog of Thompson conjecture ?

The Thompson conjecture: in a finite simple non-abelian group, there exists a conjugacy class such that every element of the group can be expressed as a product of two elements from that conjugacy ...

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### Tensor power of the natural representation of Sn

The symmetric group $S_n$ acts over $V=\mathbb{R}^n$ by permuting the canonical basis.
So it acts over $V^{\otimes p}$ with a diagonal action (acts the same over each element of the tensor product).
...

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### How to get Latin squares from a finite group and a subgroup

Let G be a finite group and we know its group table is a Latin square of order |G|. Now let H be any subgroup of G of index n. Then we can form G/H which is a collection of left cosets. My question is,...

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### Characterisation of finite solvable T-group

Definition: A $T$-group is a group in which normality is a transitive relation.
Definition: A subgroup $H \leq G$ is said to be weakly normal in $G$ if for each $g\in G$, $H^g \leq N_G(H)$ implies ...

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### Can we describe the nonabelian exterior square of a finite 2-generated metabelian group?

Let $G$ be a finite 2-generated metabelian group, and let $S$ be a schur covering group, so that we have an exact sequence
$$1\rightarrow M(G)\rightarrow S\rightarrow G\rightarrow 1$$
where $M(G)$ is ...

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### Looking for root system in finite simple groups

When looking into sizes of finite simple group of "Lie type", I observed that power of $q$ is equal to number of axes in the root system of corresponding Lie group. It is also valid for Steinberg ...