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Questions tagged [finite-groups]

Questions on group theory which concern finite groups.

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The degree prime-power graph of $Suz$

Let $G$ be a finite group. Let $\mathrm{cd}(G)$ be the set of irreducible complex character degrees of $G$ and $\rho(G)$ the set of primes dividing degrees in $\mathrm{cd}(G)$. The authors define a ...
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0answers
101 views

Normal subgroups of $p$-groups

I was reading Professor Yukov Berkovich' paper "On Subgroups of Finite $p$-groups" when I stumled upon the following theorem: Let $G$ be a nonabelian $p$-group with cyclic Frattini subgroup, $|\Phi(...
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
81 views

Calculate the character degrees of a finite group $G$

Let $G$ be a finite group and $K$ be a group of order $8$. Suppose that $G/K\cong M_{12}$ where $M_{12}$ is one of the Mathieu groups. QUESTION: How to calculate the all complex character degrees ...
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1answer
132 views

How to construct groups and large dimension representations? How about faithful ones?

Below I am referring to complex representations. We know that if $G$ is a finite group with $m=(G:Z(G))$, then every irreducible representation has size at most $\sqrt{m}$. One cannot hope for this ...
12
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1answer
2k views

Number of positions of Rubik's cube grows with multiplier 13 with the distance - what are explanations and groups with similar growth pattern?

Rubik's cube and its generalizations attracts certain attention of mathematical community. It is somehow "noteworthy" that it has been proved that diameter of the Rubik's cube group is 20, i.e. ...
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0answers
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Subgroup of the internal semidirect product [on hold]

Let $H$ and $K$ be a finite groups and $G'$ be a normal subgroup of the internal semidirect product $H.\,K$. Take $\,H'=\,H \cap \,G'$ and $K'= (G' \,H) \cap \,K$. We can see easely that $\mid G' \mid ...
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2answers
298 views

Is the cohomology ring $H^*(BG,\mathbb{Z})$ generated by Euler classes?

I am interested in the classifying space $BG$ of a finite group $G$. A real representation $V$ of $G$ of dimension $r$ defines a real vector bundle over $BG$ of rank $r$. If the determinant of this ...
5
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2answers
139 views

Obtaining quiver and relations for finite p-groups

Given a finite field $K$ with $p$ elements and a finite $p$-group $G$, is there a way to obtain the quiver and relations of $KG$ with GAP (and its package QPA)? Since $KG$ is local, the quiver should ...
3
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1answer
76 views

Splitting of regular semisimple conjugacy classes in $SL_{n}(q)$

I have the following question: Consider the following two finite groups: $GL_{n}(q)$ and $SL_{n}(q)$. What I am trying to understand is the regular semisimple conjugacy classes of $SL_{n}(q)$. Now, ...
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0answers
102 views

A question on UCS p-groups(2)

This is a follow up to this question Let $G_{1}$ and $G_{2}$ be two finite UCS p-groups with the following conditions: 1- $\vert G_{1}\vert=\vert G_{2}\vert=p^{2n}$; 2- $\Phi(G_{1})\cong\Phi(G_{2})\...
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0answers
64 views

Relationship between the p-radical subgroups and the parabolics in a BN-pair generality

A theorem of Quillen says that if $G$ is a finite Chevalley group over characteristic $p$, then the poset $\mathcal{A}_p(G)$ of nontrivial elementary abelian subgroups of $G$ is homotopy equivalent (I ...
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0answers
22 views

Decomposition into irreducible of a representation of the wreath product $S_d \wr S_n$ (4)

I have to decompose some representations of $S_d \wr S_n$. I understand better and better how it works, I still have a case I don't know how to deal with. For simplicity I take $d=2$ and $n=4$. $S^{(...
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2answers
99 views

Understanding Magma issue with maximal subgroups computation

I am trying to compute the maximal subgroups of the wreath product $(\mathbb Z/10\mathbb Z)\wr S_{99}$ using Magma's algorithm for maximal subgroups, which is an implementation of an algorithm of ...
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1answer
181 views

How quickly can one compute the Hurwitz action of braid groups on finite groups?

Let $G$ be a finite group. Define the Hurwitz action of $B_{n}$ on $G^{n}$ by letting $(x_{1},...,x_{n})\sigma_{i}=(x_{1},...,x_{i}x_{i+1}x_{i}^{-1},x_{i},x_{i+2},...,x_{n})$. I wonder what algorithms ...
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1answer
241 views

first group cohomology for the standard representation of $S_n$ over $\mathbb{F}_2$

Let $g \geq 2$ be an integer and consider the symmetric group $S_n$ where $n = 2g+1$ or $n = 2g+2$ as a subgroup of the symplectic group $\mathrm{Sp}_{2g}(\mathbb{F}_2)$ via the standard ...
2
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1answer
135 views

Decomposition into irreducible of a representation of the wreath product $S_d \wr S_m$ (2)

This is a question following Decomposition into irreducible of a representation of the wreath product $S_d\wr S_n$ Let $F$ be the trivial and $S$ be the standard representations of $S_d$ (of ...
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0answers
319 views

A hard Lefschetz theorem for nilCoxeter algebras

Let $W$ be a finite Coxeter group and $\mathcal{N}(W)$ its nilCoxeter algebra (over the reals, say), as defined at https://en.wikipedia.org/wiki/Nil-Coxeter_algebra. $\mathcal{N}(W)$ has a natural ...
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0answers
116 views

Computing the class-preserving automorphism group of finite $p$-groups

Let $G$ be a finite non-abelian $p$-group, where $p$ is a prime. An automorphism $\alpha$ of $G$ is called a class-preserving if for each $x\in G$, there exists an element $g_x\in G$ such that $\alpha(...
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1answer
2k views

Order of finite unitary group

This may be an easy exercise but I am not getting it. Let $\mathbf F_q$ be a finite field with $q$ elements and $\mathbf F_{q^2}$ be its degree two extension. Define an automorphism $\sigma$ of $\...
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1answer
93 views

Reversible polynomial circuit = polynomial reversible circuit?

I asked this in cstheory.SE a week ago. Since there are no answers or comments, and since this is perhaps more about permutations than computation, I hope it is ok to cross-post here as well. My ...
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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 ...
2
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1answer
126 views

Fixed points of the automorphisms of sporadic groups

Sporadic groups have very few outer automorphisms (in fact, $|\mathrm{Out}(G)|\leqslant2$), so it is very natural to ask what are the fixed points subgroups. For a group of Lie type (and a suitable ...
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0answers
129 views

Metrics on finite groups and generalizations of central limit theorems for balls volumes (à la Diaconis-Graham)

In wonderful lectures by P. Diaconis "Group representations in probability and statistics, Chapter 6. Metrics on Groups, and Their Statistical Use" metrics on permutation groups are considered and ...
6
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1answer
172 views

Why is Nagao's theorem the “Module theoretic version of Brauer's second main theorem”?

Let $G$ be a finite group, $p\in\mathbb{P}$ a prime, $\mathbb{F}$ an algebraically closed field of characteristic $p$, and $D\leq G$ a $p$-subgroup. Brauer second main theorem states If $\chi\in ...
12
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1answer
277 views

Genus of Cayley graph of $A_5$ with two generators of order 5

The Cayley graph of $A_5$ with two generators of order 5 seems rather complicated. I am wondering what is its graph genus (orientable or non-orientable). The best I could get by trial and error is an ...
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4answers
935 views

Is there a Cayley graph of a non-abelian finite group that is not isomorphic to any Cayley graph of any abelian group?

It's the first question I post here :) I'm sorry if the question is too specific or if it's somehow repeating others. In other words, my question is the following. Consider a Cayley graph $\Gamma$ of ...
6
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2answers
381 views

How hard is it to compute the diameter and the growth function of a finite permutation group of small degree?

Let $G \leq {\rm S}_n$ be a finite permutation group, and let $S = \{g_1, \dots, g_k\}$ be a generating set for $G$ which is closed under inversion and which does not contain the identity. The growth ...
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61 views

$2$-power-torsion elements of a group

Let $G$ be a finite group, let $P$ be one of its $2$-sylow subgroups. Let $H$ be a proper subgroup of $P$, namely $H<P$ with $H\neq P$. Is it possible that $$\bigcup_{g\in G}g^{-1}Hg=\bigcup_{g\in ...
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2answers
537 views

Characterization of Frobenius complements

I have learned that Frobenius complements are characterized (among finite groups) by having a fixed point free complex representation. That is, a finite group $G$ is a Frobenius complement if and only ...
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0answers
52 views

Efficient implementation of the Clifford group for $n$ qubits

I'm looking for an efficient implementation of the Clifford group $\mathcal{C}_n$ of $n$ qubits. The Clifford group $\mathcal{C}_n$ has stucture $(2_+^{1+2n} \circ C_8).Sp(2,n)$, where $2_+^{1+2n}$ ...
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1answer
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class structure constants relation

Let $C_{j,k}^l$ ,usually called class structure constants, eg Jansen and Boon and/or JQ Chen, be the number of times the class $l$ is generated from the product of classes $j,k$ and $c_j=c_{-j}$ (a ...
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0answers
97 views

Permutation groups with diameter $O(n \log n)$

I suspect that many permutation puzzles can be solved in $O(n \log n)$ moves, which has led me to the following question/conjecture: Suppose that 1. $P_i$ for $i<k=O(1)$ are permutations on an $n$ ...
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1answer
1k views

What finite simple groups we can obtain using octonions?

Rearranged on 2017-05-31 What I am missing is a uniform definition of finite simple groups. Especially sporadic groups are difficult to define. Many of the finite groups are defined using machinery ...
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2answers
690 views

Spherical building of an exceptional group of Lie type

I've read that one of Tits' original motivations for studying buildings was that he wanted to give a unified description of algebraic groups that would allow the definition of exceptional groups such ...
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0answers
259 views

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 ...
2
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1answer
235 views

An epimorphism into a profinite group

Let $p$ be an odd prime number, $G$ a finitely generated nonabelian profinite group, $L \lhd_o G$ a pro-$p$ group with $[G : L] = 2$. Suppose that there is a continuous surjection from $G$ onto a free ...
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1answer
222 views

Lie algebra of a p-group

Given a p-group P, the first hochschild cohomology of the group algebra (over a field of characteristic p) of P is a nonzero Lie algebra. Is it known what Lie algebra results depending on P? I have no ...
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0answers
253 views

Symmetry group and irreducible representation

Let $S$ be a bounded geometric shape in the Euclidean space $E=\mathbb{R^n}$. Assume that the origin of $E$ is a fixed point of every element of the symmetry group $G(S)$ of $S$, and assume that $G(S) ...
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1answer
194 views

Nonabelian finite groups with “locally commuting” presentation

Let $G = \left\langle S | R \right\rangle$ be a finitely presented group where S is a set of generators and R is a set of relations. We say that the presentation is "locally commuting" if whenever two ...
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1answer
188 views

Decomposition into irreducible of a representation of the wreath product $S_d\wr S_n$

Let $S_d, S_n$ be the permutation groups of $d,n$ elements. An intuitive representation of the wreath product $S_d\wr S_n$ is $V_1\otimes...\otimes V_n$, where each $V_i$ is of dimension $d$. Writing ...
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3answers
278 views

Finite groups with few conjugacy classes of maximal subgroups

Let $c$ be a positive integer, $G$ a finite group with at most $c$ conjugacy classes of maximal subgroup. What can we say about $G$? Same question, but this time $G$ is a finite group with at most $c$...
8
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1answer
226 views

Fixed-point-free group action on a finite, contractible, 3-dimensional simplicial complex

Let $K$ be a finite simplicial complex with an admissible action of a finite group $G$. (Terminology: By an action of a group $G$ on $K$ I mean an action by simplicial automorphisms. The action is ...
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1answer
367 views

Simple groups of the same order

I heard that there are no 3 nonisomorphic simple groups of the same order. Question: Is there an elementary proof of this? In case this is not the case, here a modified question: Question: Is ...
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1answer
269 views

Does the union of all finite groups yield a complete knot invariant for prime knots?

It is established in Whitten - Knot complements and groups together with the Gordon-Luecke theorem (that knot complements determine knot type) that the type of a prime knot is determined by the ...
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2answers
849 views

How does one compute induced representations for modular representations?

The set-up is this: Let $G$ be a finite group, and $H$ a subgroup. We are given an irreducible representation of $H, \rho: H\rightarrow GL_n(K)$ (I will notationally identify $\rho$ with its character)...
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0answers
80 views

On $n$th class-preserving automorphism of finite $p$-group

Let $G$ be a finite non-abelian $p$-group, where $p$ is a prime. An automorphism $\alpha$ of $G$ is called an $n$th class-preserving if for each $x\in G$, there exists an element $g_x\in \gamma_n(G)$ ...
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0answers
124 views

How can I get my hands on McKay's “Finite p-Groups” lecture notes?

The notes I'm talking about are these. I emailed Peter Cameron, but he has since moved to a different university, and has no copies himself. I also emailed the school manager at Queen Mary, but they ...
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0answers
253 views

Is the norm element characteristic in modular group rings?

Let $G$ be a finite $p$- group and let $\varphi$ be an automorphism of $\mathbb{F}_pG$ as $\mathbb{F}_p$-algebras and let $n = \sum_{g\in G} g$ be the norm element. Does it follow that $\varphi(n)=n$? ...
12
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2answers
513 views

Factorizable groups

Definition. A finite group $G$ is factorizable if for any positive integer numbers $a,b$ with $ab=|G|$ there are subsets $A,B\subset G$ of cardinality $|A|=a$ and $|B|=b$ such that $AB=G$. Problem ...