# Questions tagged [finite-groups]

Questions on group theory which concern finite groups.

1,488 questions
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### Order of unipotent matrices over $\mathbb{Z}/q\mathbb{Z}$

Let $q$ be a prime power and let $n\geq2$ be an integer. Is it known what is the largest order of a unipotent upper-triangular $n\times n$ matrix over the ring $\mathbb{Z}/q\mathbb{Z}$? I am mostly ...
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### Finite group and cyclic cover

Suppose the ﬁnite group $N$ surjects to ﬁnite group $F$. It is true that for any $G = N ⋊_α \mathbb{Z}$ there are inﬁnitely many covers of $G$ that are cyclic and surject to $F$. But is this ...
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### Group isomorphism $R^{\times} \simeq C_n\times C_2$? [closed]

Let $n$ be an positive integer and $R:=\mathbb{Z}[X]/(X^2,nX)$. Do we have $R^{\times} \simeq C_n\times C_2$ (the group of units of $R$) as it seems to be the case for small values of $n$. If so, do ...
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### What did Frobenius prove about $M_{12}$?

I am interested in this paper which I can't read because it's in German: Frobenius, G., Über die Charaktere der mehrfach transitiven Gruppen., Berl. Ber. 1904, 558-571 (1904). ZBL35.0154.02. A free ...
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### Random walk on a finite group, converging modulo a function

Let $G$ be a finite group, and let $Q$ be a probability measure on $G$. Suppose that $Q$, as a function on $G$, is supported on a conjugacy class $C$. We denote by $Q^{*k}$ the $k$-fold convolution of ...
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### Alternating Hurwitz quotients multiplicity

How many times is each alternating group a hurwitz quotient? In other words, how many non-isomorphic ways can an alternating group be generated by an element of order 2 and an element of order 3 whose ...
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### Symplectic group over finite field and quaternions

Symplectic group over finite field is defined as group preserving non-degenerate antisymmetric bilinear form on $\mathbb F_q^{2n}$. How could we define this group using quaternions ? This should be ...
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### Is there any nice way to compute transfer homomorphism in a $p$-group?

Let $G$ be a $p$-group and $H$ be a subgroup of $G$. Set $V$ be a pretransfer map from $G$ to $H$. That is, $$V(g)=\prod_{t\in T} tg(t.g)^{-1}=\prod_{t\in T_0}tg^n_tt^{-1}$$ Can we say that the ...
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### Computing the inner automorphism group of a finite Lie algebra

I'm interested in writing GAP code to compute the inner automorphism group of a finite Lie algebra. (I'd like to be able to group conjugate subalgebras together.) I've had trouble finding good ...
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### Are descents in alternating subgroup counted by $h$-vector?

Consider the alternating subgroup $A_n$ of the symmetric group $S_n$ (or in general any Coxeter Group). Is there a simplicial complex whose $h$-vector $h_i$ equals the number of elements of $A_n$ with ...
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### Finite simple subgroups of $\mathrm{GL}_n(\mathbb{C})$

Let $n$ be a fixed positive integer. Is it true that there are only finitely many isomorphism classes of finite nonabelian simple subgroups of $\mathrm{GL}_n(\mathbb{C})$? I'm especially interested ...
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### Recognition of finite simple groups by number of Sylow p-subgroups (2)

This is a follow-up to this question. Let $G$ and $G'$ be two finite simple groups of the following structures: 1- $A_{p}$, for some primes $p$; 2- $PSL_{p}(q)$, for some prime $p$ and some prime ...
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### Reference request: Models of cuspidal representations of GL(n,k) where k is a finite field

Let $k=\mathbb{F}_q$ where $q$ is a prime power of odd cardinality. Where could I find explicit models of all irreducible cuspidal (complex) representations of $GL_n(k)$ for $n\ge 3$? I understand ...
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### Low-dimensional irreducible 2-modular representations of the symmetric group

I apologize if this question is a little too basic for MathOverflow, but it's somewhat outside of my background and I'm frustrated that the answer doesn't seem to be explicit in the literature even ...
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### Maximal subgroups of a finite simple group which have some minimal subgroups in common

Let $G$ be a finite simple group and $M$ and $M'$ be two maximal subgroups of $G$. Also let $m_{M}$ be the set of minimal subgroups of $M$ and similarly $m_{M'}$ be the set of minimal subgroups of $M'$...
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### Recognition of finite simple groups by number of Sylow p-subgroups

Let $G$ and $G'$ be two finite simple groups and $p$ be a prime divisor of $\vert G\vert$ and $\vert G'\vert$. Also suppose that every Sylow p-subgroup of $G$ and $G'$ is a prime order subgroup($C_{p}$...
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### Do actions of BS(1,n) on finite sets factor through abelian quotients?

Suppose $BS(1,n)$ is the Baumslag-Solitar group and $S_m$ is the symmetric group. If $\Phi: BS(1,n) \to S_m$ is a homomorphism, must the image of $\Phi$ be abelian?
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### Extensions of a projective special linear group

Is it possible to classify groups that can be decomposed as a semidirect product $G=\mathrm{PSL}_2(q) \rtimes \langle t\rangle$, such that $t\in G$ has order $4$ and $Z(G)=\langle t^2\rangle$?
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### Intersections of products of Sylow $p$-subgroups

Motivated by the work of Cohn, Umans and collaborators on bounding the complexity of matrix multiplication, a student and I have been investigating the following problem. For subsets $X$ and $Y$ of a ...
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### Algebra for the Baby

I am reading the following article. Ryba, Alexander J.E., A natural invariant algebra for the Baby Monster group., J. Group Theory 10, No. 1, 55-69 (2007). ZBL1228.20012.. Author works with 4370-...
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### Number of generators for the Schur multiplier of a finite group

Let $G$ be a finite group, and let $M(G)=H_2(G,\mathbb{Z})$ be its Schur multiplier. Are there any known bounds on the number of generators of $M(G)$ in terms of $G$? For example, if $G$ is abelian of ...