Questions tagged [finite-groups]

Questions on group theory which concern finite groups.

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3
votes
1answer
63 views

Is the Singer cycle preserved by field automorphisms and graph automorphisms?

Let $T=\operatorname{PSL}_n(q)$ with $n$ a prime number. Then the $\mathscr{C}_3$ subgroup $M=\langle x\rangle{:}\langle\sigma\rangle$ of $T$ is isomorphic to $\mathbb{Z}_{\frac{q^n-1}{(q-1)(n,q-1)}}{:...
6
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2answers
105 views

Minimal generation of simple groups and Ore's conjecture

The well known Ore's conjecture (now established) states that every element of a finite non-abelian simple group $G$ is a commutator of a pair of elements. Also we know that $G$ is $2$-generated. I ...
1
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0answers
79 views

$G/F(G)$ is isomorphic to $X_1\times\cdots\times X_t$

I asked this question on math.stackexchange many hours ago, but haven’t got an answer. It was mentioned in a comment that the answer to my question is trivial, but I couldn’t see why. $G$ is a finite ...
1
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1answer
72 views

Is the solvable radical of a finite perfect group contained in the Schur multiplier of the quotient of the group modulo the solvable radical?

Let $G$ be a finite perfect group, and let $N$ be the solvable radical of $G$. If $G/N$ is a non-abelian simple group, then is it true that $N$ is contained in the Schur multiplier of $G/N$? If this ...
5
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1answer
164 views

Finite group ${\rm Sp}_4({\Bbb F}_3)$: involutions coming from a 4-dimensional complex representation

I am interested in the finite unitary reflection group $G= G_{32}$, the group No. 32 in Table VII on page 301 of the paper: Shephard, G. C.; Todd, J., A. Finite unitary reflection groups. Canad. J. ...
2
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1answer
92 views

How is Harish-Chandra restriction compatible with Harish-Chandra series?

Suppose $G$ is a connected reductive group over $\overline{\mathbb{F}}_p$, with Frobenius $F$. Let $(L_0,\Lambda_0)$ be a cuspidal pair with $L_0$ a Levi subgroup of a Levi subgroup $L$, and let ${^{\...
5
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0answers
63 views

Galois groups of special polynomials

This question is motivated by long experiments with GAP. Call a monic polynomial with integer coefficients special in case it is irreducible and has only coefficients $-1$, $0$ or $1$. Let $n \geq 5$....
6
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2answers
309 views

Representation of central extension

Let $G$ be a finite abelian group of rank $n$ and $H\rightarrow G$ a central extension with cyclic finite kernel. Is it true that we can find a faithful representation $H\rightarrow {\rm GL}_{k(n)}(\...
3
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0answers
90 views

Are there perfect DTI-groups which are not simple?

Recall that a subgroup $H$ of a group $G$ is called a TI-subgroup if for every $g \in G$ it is $H \cap H^g \in \{1,H\}$. We say that a group $G$ is a DTI-group if the derived subgroups of all of its ...
1
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0answers
112 views

What can we get from an automorphism of order $2$

An automorphism of order $2$ is an automorphism fixes some elements and inverts the others. It’s well known that not all groups have automorphisms of order $2$, $C_2$, for example. But if a group ...
4
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0answers
109 views

Number of elements in $\mathrm{GL}(n,p)$ with maximal order

I learned reading this question that $\mathrm{GL}(n,p)$ elements have at most a multiplicative order of $p^n -1$. I would like to know how many matrices have an order of exactly $p^n -1$. Do they ...
6
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1answer
162 views

Maximum size of $k$-Sidon set over $\mathbb{F}_2^n.$

Fix $k \in \mathbb{N}$, $k \ge 2.$ Does there exist a subset $A \subset \mathbb{F}_2^n$ such that $|A| \ge c 2^{n/k}$ with some absolutely positive constant $c,$ and satisfying $$ a_1 + a_2 + \...
1
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1answer
70 views

Outer automorphism group of $F(G)$

By a nice helpful comment in my last question, I see that if $\Phi(G)=1$ then ${\rm Out}(F(G))$($\cong G/F(G))$) is isomorphic to a direct product of ${\rm GL}(n_i,p_i)$. Actually, I’m digesting an ...
0
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0answers
60 views

Fitting subgroup of a solvable group is a direct product of some elementary abelian $p$-groups

I saw a remark saying If $G$ is solvable, then ${\rm Out}(F(G))$ is isomorphic to a direct product of ${\rm GL}(n_i,p_i)$ (except for certain value of $n_i$ and $p_i$). I know the key is to prove ...
2
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0answers
40 views

Can a polytope with vertex-transitive edge graph or face lattice be made vertex-transitive?

Let $P\subset\Bbb R^d$ be a convex, full-dimensional polytope (convex hull of finitely many points, affine hull is the whole space), $G_P$ its edge graph and $\mathcal F_P$ its face lattice. Any of ...
2
votes
1answer
200 views

Why $G/F(G)$ is isomorphic to a subgroup of ${\rm Out}(F(G))$?

I know two facts and I’ve managed to figure out how to prove one, but the other one is still a little confusing. Let $G$ be a finite solvable group and $F(G)$ is the Fitting subgroup of $G$. (...
6
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1answer
154 views

Why do we say the Fitting subgroup/generalized Fitting subgroup control the structure of a group?

I’m learning the Fitting subgroup these days. I’m interested in this topic and particularly in the role that it plays in the structure of groups. Many people on MSE mentioned that the Fitting subgroup/...
3
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0answers
53 views

Isomorphism of certain irreducible representations over finite fields

We are given a faithful representation of a cyclic group of order 5 $\rho: C_5=G \rightarrow End_{\mathbb{F}_3}(V) $ with $dim_{\mathbb{F}_3}V=8$ as vector space. It is also known that $V=U\oplus W$ ...
2
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0answers
45 views

Dimension bound on invertible bimodules for blocks

A co-author and I have recently proved the following result: Theorem. Let $B$ be a block of a finite group defined over some algebraically closed field $k$ of characteristic $p>0$. If $M$ is a $B$-...
5
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2answers
262 views

Transposition Cayley graphs are planar

Consider the Cayley graph $G$ with vertex set the elements of the symmetric group $S_n$ and generating set the set of minimal transposition generators of the group $S_n$, that is the set $S=\{(12),(13)...
20
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1answer
594 views

Is $A_5$ the only finite simple group with only 4 distinct sizes of orbits under the action of the automorphism group?

Given a finite group $G$, let $\eta(G)$ denote the number of distinct sizes of orbits on $G$ under the action of ${\rm Aut}(G)$. It happens that there are infinitely many non-abelian finite simple ...
0
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0answers
70 views

What do Sylow 2-subgroups look like for Schur covering groups of finite simple groups?

What do Sylow 2-subgroups look like for Schur covering groups of finite simple groups? Are there any references in which we can find the stucture of Sylow 2-subgroups of Schur covering groups of ...
6
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0answers
167 views

The importance/use of socle in the theory of finite groups

I asked this question in MSE, but I am not able to delete it now. It’s a little opinion-based, so it may be more suitable on this site. Definition. The socle of a group $G$, denoted ${\rm Soc}(G)$, ...
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0answers
53 views

Reference request for finite simple exceptional group of lie type $E_7(q)$ and its Schur covering group $2.E_7(q)$?

Does anyone have the paper named 'Génerateurs, relations et revêtements de groupes algébriques' written by Robert Steinberg in 1962, or any other reference for simple groups of Lie type $E_7(q)$ and ...
5
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0answers
64 views

cuspidal unipotent representation in small characteristic

Let $\mathbb{F}_q$ be a finite field with $q=p^r$ and $p$ prime. Let $G$ be a connected reductive group over $\mathbb{F}_q$. Is there a difference between the theory of unipotent cuspidal ...
2
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0answers
74 views

Dimension of center of $k[G]/\mathrm{rad}k[G]$ when characteristic of $k$ divides the order of $G$

Let $G$ be a finite group and consider $k[G]$ where $k$ is a field. In the scenario where $\mathrm{char}(k)$ divides $|G|$, how can one show that the dimension of $Z(k[G]/\operatorname{rad}k[G])$ is ...
2
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0answers
58 views

Is there a method to find the order of a lift of an element of order 2 to the Schur cover?

Let $G$ be a finite non-abelian simple group, $M.G$ the Schur covering group of $G$. Is there a method to find the order of a preimage of an element of order 2 in the natural homomorphism $\pi: M.G\...
7
votes
1answer
260 views

Is $J_1$ a subquotient of the monster group?

Edit: I was able to make a 3D diagram of the happy family if anyone is interested! https://www.youtube.com/watch?v=_4IjnIcECoQ I'm working on a twitter thread about the monster group, because I saw ...
1
vote
1answer
106 views

Examples of 3-transitive expander family of Schreier graphs

What are examples of expander family of 3-transitive Schreier graphs? Meaning for an action that is 3-transitive. It is better to have an option for randomization. We know that choosing 2 elements ...
7
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0answers
105 views

Finite groups such that non-central, commuting elements have the same stabilizer

Let me say that a finite, non-abelian group $G$ has the property $(P)$ if for any two elements $x, \, y \in G - Z(G)$ such that $y \in C_G(x)$, one has $C_G(y)=C_G(x)$. For instance, it is easy ...
2
votes
2answers
187 views

In which books we can find structure information for finite simple groups and their Schur covering groups?

In which books we can find representations or character tables, Sylow 2-subgroups and conjugacy classes for finite simple groups and their Schur covering groups and properties for Schur multiplier of ...
1
vote
1answer
158 views

Average size of iterated sumset modulo $p-1$,

Given a prime $p$, what is the average size of the iterated sumset, $|kA|$, modulo $p-1$, with $p$ a prime, and $k$ given, with $A$ chosen at random? You can pick any type of prime you like for $p$, ...
0
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0answers
101 views

Normaliser of the field automorphism of $\operatorname{GL}_n(p^f)$ in $\operatorname{GL}_{fn}(p)$

I have asked the same question on MSE several days ago here but there was no any useful comments. So I try to ask here. Let $G=\operatorname{GL}_{fn}(p)$, where $p$ is a prime and $fn>2$, and $H$ ...
1
vote
1answer
61 views

What do conjugacy classes of involutions like in finite simple group $E_7(q)$?

Are there any refences for conjugacy classes of involutions in finite simple group $E_7(q)$?
3
votes
1answer
40 views

describing embedding $U_3(q)<O_6^-(q)$, $q$ even

Let $q=2^k$. I need to explicitly construct $U_3(q)$ as a subgroup of $G=GO_6^-(q)$. It is well-known that $G\cong U_4(q)$, and as a subgroup of the latter one has $U_3(q)$ fixing a non-isotropic ...
4
votes
2answers
284 views

Is $1\neq a\in Z(2.E_7(q))\cong Z_2$ a square element in $2.E_7(q)$?

When $q$ is a power of some odd prime, is $1\neq a\in Z(2.E_7(q))\cong Z_2$ a square element in $2.E_7(q)$? A Lie algebra is a vector space $L$ over a field $K$ on which a product operation $[xy]$ is ...
5
votes
1answer
103 views

How to find a finite splitting field $K$ for $G$ such that every indecomposable $KG$-module is absolutely indecomposable

Let $G$ be a finite group and let $k$ be a finite field with char$(k)=p$ such that $p\mid |G|$. If $k$ is a splitting field for $G$, then, no matter which splitting field we take, after extending ...
1
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0answers
245 views

Differences between GAP and MAGMA [closed]

GAP and MAGMA are computer algebra systems. What are the objective differences between the two? Which capabilities are not shared? How do they compare on facilities for working with character tables?...
5
votes
2answers
494 views

Order of product of group elements

Let $G$ be a finite non-commutative group of order $N$, and let $x, y \in G$. Let $a$ and $b$ be the orders of $x$ and $y$, respectively. Can we say anything non-trivial about the order of $xy$ in ...
2
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0answers
140 views

On automorphism group

Let $p$ be a prime number, and let $\mathbb{F}_{p}$ be a finite field of order $p$. Let $U_{n}$ denote the unitriangular group of $n\times n$ upper triangular matrices with ones on the diagonal, over $...
1
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0answers
41 views

Restriction of real irreducible 2-Brauer characters to subnormal subgroups

Question: Find a finite group $G$, a subnormal subgroup $S$ of $G$, a real-valued irreducible $2$-Brauer character $\chi$ of $G$ and a real-valued irreducible $2$-Brauer character $\mu$ of $S$ such ...
20
votes
3answers
2k views

What is the geometric shape of the Monster sporadic group?

Conway made the comment that the Monster group represents the symmetries of a shape in 196,883 dimensions, something like a "star you hang on a Christmas tree." My question is, What do we know (or ...
1
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0answers
53 views

Finite $p$-groups of co-class $3$, class at least $4$ and some controlled generator growth

I am trying to prove the following comment (Ref. https://link.springer.com/article/10.1007/s00605-016-0938-5 Page-684, Rmk3.2): Let $G$ be a finite $p$-group of co-class $3$, class $\geq 4$. Then $G$ ...
7
votes
2answers
304 views

How does the Steenrod algebra act on $\mathrm{H}^\bullet(p^{1+2}_+, \mathbb{F}_p)$?

Let $p$ be an odd prime. The $\mathbb F_p$ cohomology of the cyclic group of order $p$ is well-known: $\mathrm{H}^\bullet(C_p, \mathbb F_p) = \mathbb F_p[\xi,x]$ where $\xi$ has degree 1, $x$ has ...
5
votes
1answer
108 views

How to prove the relationship between Stern's diatomic series and Lucas sequence $U_n(x,1)$ over the field GF(2)?

I found the bit count of Lucas sequence $U_n(x,1)$ over the field GF(2) is Stern's diatomic series, I want to know the reason? https://oeis.org/A002487 : Stern's diatomic series https://oeis.org/...
10
votes
1answer
177 views

Does $\chi(1)^2=|G:Z(G)|$ for irreducible character of a finite group $G$ imply $G$ is solvable?

In "Character Theory of Finite Groups" I.M. Isaacs mention the following conjecture: It is only possible in a solvable group $G$ to have $\chi(1)^2=|G:Z(G)|$ with $\chi \in$ Irr$(G)$. Is this ...
3
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0answers
355 views

Is every generator of $Z({\rm Spin}_n^{\epsilon}(q))$ a square element in ${\rm Spin}_n^{\epsilon}(q)$?

A. I wonder if every generator of $Z({\rm Spin}_n^{\epsilon}(q))$ is a square element in ${\rm Spin}_n^{\epsilon}(q)$? B. When $Z(\Omega_{2m}^{\epsilon}(q))\cong C_2$, is the unique element of order ...
6
votes
1answer
226 views

Conjugation in finite simple classical groups

Let $G_n(q)=\mathrm{(P)SL}_n(q), \mathrm{(P)SU}_n(q),\mathrm{(P)Sp}_{2n}(q)$, $\Omega_{2n+1}(q), (P)\Omega^\pm_{2n}(q)$ be a simple classical group. Consider the natural embedding $G_{n-1}(q) \subset ...
4
votes
1answer
223 views

Why do Nakajima and Watanabe claim the induced action of a finite linear group on the invariant subring of the reflection subgroup is linearizable?

I just picked up the paper "The classification of quotient singularities which are complete intersections" by Haruhisa Nakajima and Kei-Ichi Watanabe, which is in the book Greco, Silvio, and Rosario ...
1
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0answers
79 views

Terminology for representation all of whose isotypic pieces are nontrivial

Let $V$ be a finite-dimensional representation of a finite group $G$. Is there an adjective describing those $V$ for which every irreducible representation of $G$ is a direct summand of $V$?

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