# Questions tagged [finite-groups]

Questions on group theory which concern finite groups.

**0**

votes

**0**answers

25 views

### 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 ...

**0**

votes

**0**answers

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**

votes

**3**answers

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:
...

**0**

votes

**0**answers

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 ...

**3**

votes

**1**answer

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**

votes

**1**answer

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. ...

**-2**

votes

**0**answers

44 views

### 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
...

**9**

votes

**2**answers

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**

votes

**2**answers

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**

votes

**1**answer

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, ...

**1**

vote

**0**answers

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})\...

**5**

votes

**0**answers

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 ...

**2**

votes

**0**answers

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^{(...

**2**

votes

**2**answers

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 ...

**1**

vote

**1**answer

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 ...

**6**

votes

**1**answer

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**

votes

**1**answer

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 ...

**13**

votes

**0**answers

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 ...

**1**

vote

**0**answers

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(...

**13**

votes

**1**answer

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 $\...

**2**

votes

**1**answer

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 ...

**12**

votes

**0**answers

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**

votes

**1**answer

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 ...

**1**

vote

**0**answers

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**

votes

**1**answer

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**

votes

**1**answer

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 ...

**11**

votes

**4**answers

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**

votes

**2**answers

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 ...

**0**

votes

**0**answers

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 ...

**9**

votes

**2**answers

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 ...

**3**

votes

**0**answers

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 $2_+^{1+2n}$ ...

**-1**

votes

**1**answer

111 views

### 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 ...

**8**

votes

**0**answers

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 $n$ ...

**21**

votes

**1**answer

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 ...

**10**

votes

**2**answers

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 ...

**2**

votes

**0**answers

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**

votes

**1**answer

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 ...

**6**

votes

**1**answer

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 ...

**6**

votes

**0**answers

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) ...

**8**

votes

**1**answer

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 ...

**3**

votes

**1**answer

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 ...

**11**

votes

**3**answers

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**

votes

**1**answer

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 ...

**7**

votes

**1**answer

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 ...

**9**

votes

**1**answer

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 ...

**7**

votes

**2**answers

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)...

**1**

vote

**0**answers

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 $g_x\in \gamma_n(G)$ ...

**3**

votes

**0**answers

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 ...

**11**

votes

**0**answers

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**

votes

**2**answers

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 ...