# Questions tagged [finite-groups]

Use with the (group-theory) tag. The tag "finite-groups" refers to questions asked in the field of Group Theory which, in particular, focus on the groups of finite order.

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### Can Erdős-Turán $\frac{5}{8}$ theorem be generalised that way?

Suppose for an arbitrary group word $w$ ower the alphabet of $n$ symbols $\mathfrak{U_w}$ is a variety of all groups $G$, that satisfy an identity $\forall a_1, … , a_n \in G$ $w(a_1, … , a_n) = e$. ...
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### Induced representation, Ind(Res(U))

I am reading a book of Fulton and Harris "Representation theory, a first course". Now it's all about representation theory of finite groups, and there is one exercise, which I can't solve: ...
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### The division algebras arising in the Wedderburn decomposition of a finite group modulo its radical in characteristic $p$

The following question is probably straightforward for those who know. However, I am used to working either over splitting fields or in characteristic zero. Question. Let $G$ be a finite ...
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### Infinite family of finite groups without surjections

Is there an infinite family $\mathcal{F}$ of finite groups such that their exponent is bounded, i.e. $\exists N \geq 1 : \forall x \in G \in \mathcal{F}, x^N = 1_G$; there does not exist any ...
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### Why do we trust the Classification of Finite Simple Groups?

It seems to me there are a two main reasons to believe a theorem/conjecture to be true: Because it has a correct proof (e.g. the Feit-Thompson Theorem, Dirichlet's Theorem) Because there is an ...
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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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### Small group characterizing identity matrix

I am looking for a small (say, finite and of small cardinality) subgroup of the general linear group whose centralizer consists only of scalar matrices. I work over complex numbers. A more precise ...
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### Does asymmetric fraction of finite groups tend to $0$?

Let’s define asymmetric fraction of a finite group $G$ as the number $af(G) = \frac{|\{(g, a) \in G \times Aut(G)| a(g) = g\}|}{|G||Aut(G)|}$. Equivalently it can be defined as $P(A(X) = X)$, where $A$...
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### Is the matrix $\sum_{g \in G} a_g \rho(g)$ normal and what further properties does it have?

Let $\rho$ be the regular representation of $G$. $S \subset G$ a generating set, $|g| := |g|_S=$ word length with respect to $S$. Then I construct such a matrix, where we have some ordering $g_i$ of ...
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### Upper Bound Lemma implies the Ergodic Theorem for Random Walks on Groups?

Cross posted on Mathoverflow 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 proper ...
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### Slick proof that Coxeter groups $H_3$ and $H_4$ are finite?

I'm planning to teach a course on reflection/Coxeter groups next Fall, and have started outlining the first days, which will presumably be the classification of finite reflection groups. Here is an ...
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### Restricting irreps of $S_n$ to $D_n$ of order $2 n$

I would like to know how to restrict the irreps of the symmetric group $S_n$ to the dihedral group $D_n$ of order $2 n$. We know that $D_n < S_n$. Symmetric group $S_n$ Due to Hardy and Ramanujan ...
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### Why is studying centralizers the/a key to classifying finite groups?

In this MO thread https://mathoverflow.net/questions/38161/heuristic-argument-that-finite-simple-groups-ought-to-be-classifiable, Borcherds says One problem, as least with the current methods of ...
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### Inverse Galois problem

The inverse Galois problem conjectures that every finite group is (isomorphic to) the Galois group of some Galois extension of $\mathbb Q$, however it is not known. My question is: what is the ...
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### Complexity of testing if a binary operation is a group

Given a binary operation specified as an $n \times n$ Cayley table, what is the complexity of the best deterministic algorithm for testing if the binary operation is a group? There's a fairly simple ...
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### Reference request: groups of order $p^4$.

I am looking for a textbook or a paper which include the classification of groups of order $p^4$ ($p$ is prime) using generators and relations. In particular I like to understand which group $G$ "...
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### Missing $\{2,p\}$-Hall subgroups in finite non-abelian simple groups

Can anybody tell me how to prove this theorem? Theorem: Suppose that $G$ is a finite non-abelian simple group. Then there exists an odd prime $p\in\pi(G)$ such that $G$ has no $\{2,p\}$-Hall ...
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### an interesting coincidence with group orders: is there some theorem explaining it?

The sequence OEIS A333646: $1,6,15,28,30,33,40,42,51,66,69,84,91,95,102,105,117,120,135,138,140,141,145,159,165,182,186,190,210,213,\cdots$ Contains numbers that are divisible by the largest prime ...
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### Let $G$ be a nilpotent group generated by a finite set of torsion (i.e. finite order) elements. Show that $G$ is finite.

Given: Let $G$ be a nilpotent group generated by a finite set of torsion (i.e. finite order) elements. Show that $G$ is finite. Also would love to know if it's possible to show that an infinite ...
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### Proof of a Burnside theorem without character theory?

Burnside proved, by use of character theory, that if a finite group $G$ has a conjugacy class $C$ such that $\vert C \vert$ is a prime power $> 1$, then $G$ is not simple. Let us call this ...
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### Historical Question about Schur-Zassenhaus Theorem

I couldn't find any historical information about Schur-Zassenhaus theorem in many books or even papers which mention this theorem. I think, Schur proved that if $G$ is a finite group and if $N$ is ...
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### Discrete subgroups of SU(n) and SO(n).

Thank you very much for your concern. I am in physics background, any simpler but complete explanation would be helpful. I would like to know whether there is a complete understanding of discrete ...
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### Fixed points of coset operation

Let $G$ be a finite group which operates on two finite sets $E_1$ and $E_2$. Say that $E_1$ and $E_2$ are weakly $G$-isomorphic if for every $g \in G$, $\mathrm{Card}(E_1^g)=\mathrm{Card}(E_2^g)$, ...
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Suppose G has a cyclic normal subgroup $\langle a\rangle$ of order $m$ and prime power index $s$ such that $m$ and $s$ are relatively prime. Then the following exact sequence splits: 1 \...