Questions on group theory which concern finite groups.

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2
votes
1answer
103 views

Symplectic group over integers and finite fields

For $H=\left( \begin{smallmatrix} 0 & I_n \\ -I_n & 0 \end{smallmatrix} \right)$ and a commutative ring $F$, the symplectic group $Sp(2n,F)$ is the set of all matrices $M\in F^{2n\times 2n}$ ...
-5
votes
0answers
13 views

applications of systems of linear equations [on hold]

A person plans to invest a total of ​$260,000 in a money market​ account, a bond​ fund, an international stock​ fund, and a domestic stock fund. She wants 60​% of her investment to be conservative​ ...
1
vote
0answers
67 views

p-groups with unique normal minimal subgroup

Is p-groups with unique normal minimal subgroup have been Classification? Is there any article on the subject?
0
votes
1answer
56 views

About $c(A)$ in $c(A)|A|\leq |A^{-1}A|$

Let $G$ be a finite group, $\emptyset\neq A\subseteq G$, $A^{-1}:=\{ a^{-1}:a\in A\}$, and put $$c(A):=\max\{t\in \mathbb{Z}: t|A|\leq |A^{-1}A|\}$$ It is clear that $1\leq c(A)\leq ...
1
vote
0answers
57 views

Computing abelianizations of some explict finite subgroups of $GL_2(\mathbb{Z}/p^n \mathbb{Z})$

I have been attempting to find some abelianizations of some subgroups of $GL_2(\mathbb{Z}/p^n \mathbb{Z})$. I have been using brute force for the most part but I get very messy results. Here is an ...
11
votes
1answer
209 views

The Weyl group of E8 versus $O_8^+(2)$

Right now Wikipedia says: The Weyl group of $\mathrm{E}_8$ is of order 696729600, and can be described as $\mathrm{O}^+_8(2)$. The second part feels wrong to me. $\mathrm{O}^+_8(2)$ is the ...
11
votes
1answer
242 views

A sum over characters of the symmetric group

Let $C_\mu$ be the size of the conjugacy class in $S_n$ of permutations whose cycletype is the partition $\mu\vdash n$. Let $\chi$ be the characters of the irreducible representations of $S_n$. Let ...
-4
votes
1answer
118 views

Even-odd partitioned groups! [closed]

Let $G$ be a group with the property $G=G_e\dot{\cup} G_o$ with $G_oG_o\subseteq G_e\leq G$. ($\dot{\cup}$ denotes disjoint union, $\leq$ is subgroup notation, and $G_o^{-1}=\{x^{-1}: x\in G_o\}$) ...
6
votes
2answers
183 views

Uniqueness of the fusion ring for simple finite group

We know that the irreducible representations $R_i$ of a group $G$ can give rise to a fusion ring: $R_i\otimes R_j = \oplus_k N^{ij}_k R_k$. I wonder if the following statement is true or not: If $G$ ...
8
votes
1answer
174 views

Finite groups: equations with many solutions

Let $\omega$ be a word in the free group on generators $x_1,x_2,\ldots,x_n,g_1,g_2,\ldots,g_k$, where $n>0$ and $k\geq 0$. For any finite group $G$ and elements $g_1,g_2,\ldots g_k$ in $G$ we ...
6
votes
1answer
195 views

How do I determine a real matrix form for a group representation?

Hello mathoverflow community, I am a little stucked working on my master thesis. For a representation on $\mathbb{Z}_p\ltimes\mathbb{Z}_p^*$ induced from the additive character $\chi$ of ...
12
votes
1answer
124 views

An inequality for the minimal number of generators of a finite group

Let $G$ be a finite group, $n(G)$ the minimal number of generators and $m(G)$ the minimal number of irreducible complex representations generating (with $\otimes$ and $\oplus$) the left regular ...
2
votes
0answers
114 views

Groups with Abelian Automorphism Group

In a paper, the authors Jonah-Konvisser say Until recently (~1975), there were no published examples of non-abelian groups with abelian automorphism groups. Heinken and Liebeck have methods for ...
5
votes
0answers
178 views

Factorization of permutations into two factors with fixed number of cycles, plus a placement condition

In my recent work I have been led to consider the following type of permutation factorizations. Let $\pi_1$ and $\pi_2$ be two fixed permutations, with disjoint support, i.e. $\pi_1$ acts on ...
3
votes
1answer
116 views

Must normalizing field outer automorphisms “divide” the dimension?

Imprecise question: To get a normalizing field outer automorphism of order $r$, must we multiply the dimension by $r$? Precise hypothesis: Let $p\geqslant 5$ be a prime, let $q$ be a power of $p$ and ...
4
votes
1answer
189 views

The parity of the full automorphism group order of finite non-abelian groups of prime exponent

Is there a finite non-abelian group $G$ of prime exponent such that the full automorphism group of $G$ is of odd order?
0
votes
0answers
57 views

long root elements fixed by an automorphism in simple lie type group

Thanks for any help or comments. Suppose $G=G(q)$ is a simple lie type group over field $F$ of characteristic $r$. So $G$ has some well known subgroup $X_a$ named Long root subgroup such that ...
8
votes
2answers
225 views

Irreducible reps and characters of $G \rtimes A$

Is there a theorem which classifies irreducible representations of semi-direct product of finite groups $G \rtimes A$, where $A$ is a finite abelian group and hence write down the character table for ...
7
votes
1answer
267 views

For a new operation on a finite group of odd order giving a loop structure, when does this also gives a group

For finite groups $G$ of odd order, as $x \mapsto x^2$ is bijection (but no automorphism in general) then, we can define for each $g \in G$ the element $x^{1/2}$ by requiring $(x^{1/2})^2 = x$. Then ...
1
vote
0answers
63 views

On the unipotent conjugacy classes in $SU(3,q^2)$

Consider the special unitary group of degree 3 over a finite field $\mathbb{F}_{q^2}$, $q=p^n$ a prime power, and $U$ its Sylow $p$-subgroup (we way fix it to be the subgroup of upper triangular ...
2
votes
0answers
269 views

What kind of group invariants exist? [closed]

Let $G$ be a finite group.Then it is known, that: 1) The group determinant determines the group (up to isomorphism) 2) The 1, 2, 3 characters determine the group 3) The invariants $f_1,\cdots,f_m$ ...
8
votes
1answer
270 views

perfect group of order 190080

I need to know some properties of the perfect group of order 190080 which is the schur cover of Mathieu group 12 but by using PerfectGroup(190080), gap runs so slowly. Is there any other order in Gap ...
3
votes
1answer
107 views

embedding of $O_4^-(q)$ in $U_4(q)$

For $q$ an odd prime power, one can construct $H:=O_4^-(q)<U_4(q)$ by treating the $H$-invariant bilinear form as Hermitean. On the other hand $U_4(q)$ is isomorphic to $O_6^-(q)$; I need to ...
14
votes
2answers
455 views

factorization of the regular representation of the symmetric group

Let $\mathbb{C}[S_n]$ be the regular representation of the symmetric group $S_n$, and let $\mathbb{C}^n$ be the vector representation. Question: Does there exist a representation $V$ (of dimension ...
5
votes
1answer
139 views

Common zero of invariants of finite groups

Let $G$ be a finite group $n = |G|$. Let $\sigma : G \rightarrow GL(n,\mathbb{C})$ be the regular representation. Hence every element of $G$ can be seen as a permutation matrix. Let ...
4
votes
2answers
142 views

Frobenius Groups of Automorphisms

Recently, I am looking different papers on the topic $$\mbox{Frobenius groups of automorphisms of a group.}$$ But I am familiar with Frobenius groups only; not with their (faithful) actions on groups. ...
4
votes
1answer
140 views

Connection between cyclic group and exponential function

I have been thinking about this for a while, but now got to the point where I got stuck. I don't know if it might be considered as a research level question, but I would be very happy if somebody knew ...
5
votes
0answers
93 views

Restriction of $H^3(M_{24}, U(1))$ to $M_{12} \rtimes \mathbb{Z}_2$

$M_{12}\rtimes \mathbb{Z}_2$ is a maximal subgroup of $M_{24}$, where $M_{24}$ and $M_{12}$ are Mathieu groups . Also, it is known that $H^3(M_{24}, U(1)) \cong \mathbb{Z}_{12}$. I want to find the ...
-1
votes
1answer
159 views

Which finite cyclic groups can be characterized by Lattice isomorphism and isomorphism between their automorphism groups?

Given a finite cyclic group $G$, we denote by $L(G)$ the lattice of its subgroups, and by $\mathop{\rm Aut}(G)$ the automorphism group of $G$. Let $H$ be any group. Assume that $L(G)\cong L(H)$ and ...
2
votes
2answers
216 views

Maximal size of minimal generating set

Let $G$ be a finite group. Denote by $D(G)$ the maximal size of a minimal generating set, (minimal in the sense of inclusion). I vaguely remember seeing recently something on $D(G)$. Can anyone refer ...
5
votes
0answers
100 views

When is a Hermitian matrix of the form $g^*g$ for some matrix $g$

I tried asking this question on Math StackExchange and didn't get any replies. I was hoping that maybe someone here could help, sorry for duplicating. I'm trying to figure out some properties of ...
0
votes
0answers
69 views

When can a 2-cocycle on a subgroup can be extended?

This question is based on a question when is the restriction $H^2(G,\mathbb{C}^*)\to H^2(K,\mathbb{C}^*)$ surjective? I am asking this as a new question as I already asked that user but got no ...
5
votes
1answer
167 views

Applications of the Galois embedding problem

Given a finite Galois extension of number fields $L/K$ with Galois group $G$ and a surjection $E\twoheadrightarrow G$ of finite groups, the Galois embedding problem is the question of whether there ...
2
votes
0answers
71 views

Modularisation on group representations with arbitrary braiding

Applying the modularisation/deequivariantisation procedure to the representation category $\operatorname{Rep}_G$ of a finite group $G$ with trivial braiding gives the fibre functor to vector spaces. ...
11
votes
1answer
306 views

Can a large transitive permutation group need many generators?

let $G$ be a transitive permutation group acting on $\{1, \ldots, n\}$, and let $d(G)$ be the minimal number of generators of $G$. Is it true, that for $n\rightarrow\infty$ we have ...
0
votes
1answer
98 views

Perfect $Q[G]$-complex

Let $G$ be a finite group and let $M$ be a perfect $\mathbb{Q}[G]$-complex. Suppose that $M\otimes_{\mathbb{Q}[G]}\mathbb{Q}$ is quasi-isomrphic to $0$ can we conclude that $M$ is quasi-isomorphic to ...
3
votes
1answer
201 views

When the commutativity of the product of group subsets implies the commutativity of the group?

Let $G$ be a finite group. Which are the values of $k$ for which if every two $k$-subsets of $G$ commute then $G$ is commutative? Clearly, this holds for $k=1$ and $k=2$.
0
votes
0answers
48 views

Normal conjugate of elements of unipotent upper tringular matrices over F_q

Let $UT_n(q)$ be the group of upper triangular matrices with entries in the finite field $F_q$ and ones on the diagonal. Denote the normal closure of an element $s\in UT_n(q)$ by $s^{UT_n(q)}$, i.e., ...
6
votes
4answers
471 views

When does $\langle grg^{-1}|g\in G\rangle = \langle gsg^{-1}|g\in G\rangle$ imply $\langle r\rangle=\langle xsx^{-1}\rangle$?

We call a finite group $G$ normal if for all $s,r\in G$, if $\langle gsg^{-1}:g\in G\rangle =\langle grg^{-1}:g\in G\rangle $ then there exists $x\in G$ such that $\langle r\rangle =\langle ...
1
vote
1answer
121 views

Are rotational isometries groups generated by some kind of rotations?

If we consider the index $2$ subgroup of a Weyl group consisting of the isometries with determinant $1$ (the 'special' Weyl group), is it known that it is generated by rotations around some fixed ...
1
vote
1answer
58 views

Finite groups of planar homeomorphsims

Let G be a finite subgroup of the group H of orientation-preserving homeomorphisms of the plane that fix the origin. Is G conjugate in H to a group of rotations? I've been told this result was ...
0
votes
1answer
49 views

How to claculate the $T$-stable subgroup of second cohomology group

Let $G=\langle x,y,z,w \mid [y,x]=w^p=z, x^{p^2}=y^p=z^p=1 \rangle$ be a group, where $[u,v]=u^{-1}v^{-1}uv$, $p$ is a prime and the commutator which do not appear is 1. Let $N=\langle y,w \rangle ...
3
votes
1answer
213 views

Finite groups of order $n$ having exactly $n$ subgroups

Is it known a characterization of finite groups of order $n$ having exactly $n$ subgroups? A supplementary question: are there abelian groups other than the trivial group and $\mathbb{Z}_2$ with ...
5
votes
1answer
102 views

Generalization of a lemma of Livne

Let $H$ be a finite $2$-group. Let $N_{4}(H)$ be the subgroup generated by fourth powers. Let $H_{4}$ be the last term in the short exact sequence $1\rightarrow N_4(H) \rightarrow H \rightarrow H_{4} ...
13
votes
3answers
407 views

Explicit construction of an element of ${\rm GL}(2, p)$ of order $p+1$

It is well-known that the order of $GL(2, p)$ is $(p^2-1)(p^2-p) = (p-1)^2(p+1)p.$ It is easy to construct matrices of orders $(p-1)$ and $p$ (diagonal and parabolic, respectively), but the only way ...
6
votes
1answer
113 views

Generate harmonic polynomials for a finite group

Let $G$ be a finite group acting on a complex vector space $V$. Let $\mathcal{D}$ denote the differential operators with constant coefficients and $\mathcal{D}^{G}$ be the $G$-invariant operators. A ...
1
vote
1answer
126 views

The groups with nilpotent Hall $p'$ subgroup

Theorem $1$(Burnside): A simple nonabelian finite group can not have a conjugacy classes with prime power elements. Theorem $2$: A group of order $p^nq^m$ is solvable. Theorem $1$ depends on ...
10
votes
0answers
153 views

Are all sneaky groups products of Frobenius and 2-Frobenius groups?

I've been stuck thinking about this for a while. Def. Let $G$ be a finite solvable group whose order is divisible by only three primes: $p,q,$ and $r$. Suppose that $G$ has cyclic subgroups of ...
43
votes
1answer
839 views

Are there $n$ groups of order $n$ for some $n>1$?

Denote $N(n)$ : number of groups of order $n$ Does $N(n)=n$ hold for some $n>1$ ? I checked the OEIS-sequence https://oeis.org/A000001 as well as the squarefree numbers in the range ...
5
votes
1answer
155 views

Strongly real elements of odd order in sporadic finite simple groups

Recall that an element of a finite group is said to be real if it is conjugate to its inverse, and strongly real if the conjugating element can be chosen to be an involution. Question: Is it true ...