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Questions tagged [finite-groups]

Questions on group theory which concern finite groups.

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133 views

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 ...
3
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1answer
104 views

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 ...
6
votes
2answers
156 views

$p$-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 ...
4
votes
1answer
219 views

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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0answers
114 views

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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0answers
41 views

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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2answers
530 views

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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0answers
167 views

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 ...
7
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1answer
323 views

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 ...
7
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2answers
248 views

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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0answers
112 views

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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86 views

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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1answer
80 views

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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2answers
229 views

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 ...
1
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1answer
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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1answer
135 views

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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0answers
114 views

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 $G$ ...
5
votes
3answers
285 views

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 ...
12
votes
2answers
391 views

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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0answers
105 views

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\...
3
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1answer
129 views

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 $&...
1
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1answer
116 views

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 ...
5
votes
2answers
311 views

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 ...
2
votes
1answer
100 views

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}_{\...
4
votes
0answers
207 views

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 ...
27
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2answers
1k views

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 $BH^{+}$ ?
2
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0answers
127 views

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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0answers
33 views

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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0answers
175 views

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 ...
2
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1answer
224 views

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 $...
4
votes
2answers
215 views

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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0answers
22 views

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 $\Omega(P) \subset Z(P)$). ...
3
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1answer
150 views

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 ...
3
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0answers
106 views

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 ...
1
vote
1answer
175 views

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(\...
12
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0answers
175 views

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 ...
1
vote
1answer
111 views

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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votes
0answers
244 views

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-...
6
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1answer
164 views

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 $p$-...
7
votes
2answers
336 views

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 ...
12
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3answers
534 views

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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0answers
93 views

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 ...
6
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0answers
254 views

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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0answers
135 views

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 $(\...
13
votes
2answers
351 views

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 ...
13
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1answer
397 views

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). ...
2
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1answer
117 views

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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0answers
53 views

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 ...
5
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0answers
88 views

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 ...
4
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0answers
130 views

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 ...