Questions on group theory which concern finite groups.

**7**

votes

**1**answer

244 views

### Does the Fano plane “embed” in the complex projective plane?

$PSL(2,7)$ acts on the projective plane over $\mathbb{F}_2$ (the Fano plane) through its identification with $GL(3,2)$. It also acts on the projective plane over $\mathbb{C}$ through either of its ...

**0**

votes

**0**answers

107 views

### On the structure of groups according to their conjugacy classes

A finite group $G$ satisfies property $P_n$ if for every prime integer $p$, $G$ has at most $(n-1)$ non-central conjugacy classes the order of the representative element of which is a multiple of $p$. ...

**3**

votes

**0**answers

118 views

### Cocycle condition for 2-groups

I know that if $\omega_d(g_1, \ldots, g_d)$ is an d-cocycle characterized by $H^d(G,U(1))$, it satisfies the co-cycle condition
$(d\omega_d)(g_1, \ldots, g_{d+1}) = g_1.\omega_d(g_2,\ldots,g_{d+1}) + ...

**1**

vote

**1**answer

68 views

### Character kernels in the lattice of subgroups of a finite abelian group

I am looking for any efforts that have been made to characterize the character kernels (equivalently, the subgroups yielding cyclic quotients) inside the lattice of subgroups of a finite abelian group....

**7**

votes

**1**answer

307 views

### Is $[729,57]$ a Sylow $3$-subgroup of some well-known group?

Let $G$ be the group $[729,57]$, using GAP's notation. I have so far two descriptions of the group:
a presentation
an embedding (not surjective!) of the group into a Sylow $3$-subgroup of the unit ...

**1**

vote

**1**answer

193 views

### Finite groups $G$ satisfying property $P_n$

A finite group $G$ satisfies property $P_n$ if for every prime integer $p$, $G$ has at most $(n−1)$ non-central conjugacy classes the order of the representative element of which is a multiple of $p$. ...

**2**

votes

**0**answers

74 views

### Groups with special character degrees

Let $G$ be a finite group of order $p_1^{a_1}\times p_2^{a_2}\times\cdots \times p_n^{a_n}$. Is there any classification for simple groups such that for each $i$, $p_i^{a_i}$ is an irreducible ...

**4**

votes

**2**answers

93 views

### sum of squares of Schur polynomials indexed over partition valued functions on a set

Fix a finite set $X$ and two natural numbers $d$ and $n$.
For a partition $\lambda$ and a number $d$ denote by $s_\lambda^d(x_1,\dots,x_d)$ the Schur polynomial in $d$-many variables $x_1,\dots,x_d$. ...

**7**

votes

**3**answers

464 views

### Which groups are Galois over some p-adic field?

Suppose I have some finite $p$-group $G$, or a little extension of it.
How do I know if there exists a prime $l$ and a finite extension $K$ of $\mathbb{Q}_l$ such that $G$ is the Galois group of ...

**6**

votes

**1**answer

562 views

### Positivity of the alternating sum of indices for boolean interval of finite groups

Let $G$ be a finite group and $H$ a subgroup such that the interval $[H,G]$ is a boolean lattice.
Let $L_1, \dots , L_n$ be the maximal subgroups of $G$ containing $H$.
Let the alternative sum ...

**1**

vote

**1**answer

103 views

### Can the reversed lattice of a subgroups interval be represented?

Let $G$ be a finite group and $H$ a subgroup. The interval $[H,G]$ is the lattice of overgroups of $H$.
It is an open problem to know if every finite lattice can be represented by such an interval (...

**4**

votes

**1**answer

308 views

### Maximal set of non commuting elements in a conjugacy class of $S_8$

Can someone calculate by computer or prove the subset of maximal order of pairwise non commuting elements in the set of conjugacy class containing $(123)(45)$ in $S_8$?
I mean a subset of conjugacy ...

**2**

votes

**0**answers

104 views

### Generalization of the sign representation to Hopf algebras

For $G$ a finite group, the sign representation is the one-dimensional representation $\pi : G \to \mathbb{C}$ with $\pi(g)$ the sign of the permutation given by the action of $g$ on $G$ by left ...

**5**

votes

**1**answer

240 views

### Zero-sum sets in union-closed families

The Davenport constant $D(G)$ of a finite abelian group $G$ is the minimum integer $n$ such that whenever $a_1, \ldots, a_n \in G$ (not necessarily distinct), there is a non-empty $I \subseteq [n]$ ...

**1**

vote

**1**answer

101 views

### Centralizer of a central involution in a simple group of Lie type

Let $G$ be a finite simple group of Lie type and $x$ be a central involution (that is, an involution which is contained in the center of a Sylow $2$-subgroup).
Is it true that, if $y$ is another ...

**2**

votes

**1**answer

125 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 $M\in F^{2n\times 2n}$ ...

**1**

vote

**1**answer

135 views

### p-groups with unique normal minimal subgroup

Have $p$-groups with a unique normal minimal subgroup been classified? Is there any article on the subject?

**0**

votes

**1**answer

69 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 $1\leq c(A)\leq \frac{|G|}{|A|}$,...

**2**

votes

**0**answers

99 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

**1**answer

244 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 ...

**12**

votes

**1**answer

282 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 $\...

**-3**

votes

**1**answer

137 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

**2**answers

205 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

**1**answer

188 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

**1**answer

213 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 $\mathbb{Z}...

**12**

votes

**1**answer

149 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

**0**answers

140 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

**0**answers

196 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 $\{1,......

**3**

votes

**1**answer

181 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

**1**answer

204 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

**0**answers

69 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 $X_a\...

**8**

votes

**2**answers

253 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

**1**answer

275 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 ...

**2**

votes

**0**answers

277 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 $f_1,\cdots,f_m$ ...

**10**

votes

**1**answer

367 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 the Mathieu group ${\rm M}_{12}$, but
when using ...

**3**

votes

**1**answer

112 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

**2**answers

489 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

**1**answer

152 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 $\mathbb{Q}[x_1,......

**4**

votes

**2**answers

163 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

**1**answer

150 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

**0**answers

100 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

**1**answer

169 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

**2**answers

224 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

**0**answers

104 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

**0**answers

71 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

**1**answer

179 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

**1**answer

114 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

**1**answer

321 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 $\frac{d(G)\log|G|}{...

**0**

votes

**1**answer

105 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 $...

**4**

votes

**1**answer

213 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 $k=2$.