# Questions tagged [finite-groups]

Questions on group theory which concern finite groups.

1,488 questions
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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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### 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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### 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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### 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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### 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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### 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 ...