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Questions tagged [finite-groups]

Use with the (group-theory) tag. The tag "finite-groups" refers to questions asked in the field of Group Theory which, in particular, focus on the groups of finite order.

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Sum of all elements of a $p$-section of a group

Let $G$ be a finite group. A $p$-section of $G$ is a subset of the form $S(x) = \{g \in G \mid g_p \text{ is conjugate to } x\}$, where $x$ is a $p$-element of $G$, with $p$ a prime, and $g_p$ denotes ...
Gauss's user avatar
  • 2,683
3 votes
0 answers
47 views

Question about a possible application of Burnside's basis theorem

Let $G=\langle a_1,\dots , a_d\rangle$ be a $d$-generator $p$-group of order $p^n$ (i.e. $d$ is the minimal number of generators). Further let $N$ be a characteristic and elementary abelian subgroup ...
Aericura's user avatar
  • 291
6 votes
2 answers
104 views

Does an irr. rep. of finite $G$ have a basis of the form $\{hv : h \in H\}$ for $H$ subgroup of $G$?

Let $G$ be a finite group and $V$ be a (complex vector space) representation of $G$. Consider the following three facts: Whenever a $V$ is an irreducible representation of $G$, the dimension of $V$ ...
Samuel Johnston's user avatar
0 votes
1 answer
51 views

How to check whether a finite $p$-group is regular in GAP?

I am trying to check whether a given $p$-group is a regular $p$-group in GAP. I am trying to use the command 'IsRegularPGroup(G)' for it. However I am getting 'Error, Variable: 'IsRegularPGroup' must ...
cryptomaniac's user avatar
2 votes
0 answers
26 views

Small examples of non-transitive Automorphism groups of Steiner Systems

I'm currently doing research for a bachelor's seminar talk. I have found a result from E. Mendelsohn, "On the groups of automorphisms of Steiner triple and quadruple systems" stating that ...
dilemmma's user avatar
2 votes
0 answers
85 views

Brauer Tables for the Monster Group

I am doing research on modular character theory and wanted to study the Brauer Table for some of the larger simple groups, however I have been struggling to find the Brauer Tables for the monster ...
Zach M's user avatar
  • 39
4 votes
1 answer
103 views

Criterion for cyclic groups in terms of its number of subgroups

Let $G$ be a finite group of order $n$. Suppose that for every other group $H$ of order $n$, the number of subgroups of $H$ is at least as big as the number of subgroups of $G$. Does it imply that $G$ ...
sagnik chakraborty's user avatar
11 votes
1 answer
203 views

A group with exactly half of the elements in one conjugacy class

Suppose $G$ is a finite group and there exists a conjugacy class $S$ in $G$ contains exactly $|G|/2$ many elements. What can we say about $G$? For example if $G$ is a Dihedral group $D_{n}$ with $n$ ...
cybcat's user avatar
  • 786
4 votes
1 answer
49 views

Derived series of a square-free order group stabilizes

I have been studying from these notes on groups, rings and fields by Lenstra and I find myself struggling with problem 1.20 which states the following Let $G$ be a finite group of squarefree order. ...
vhis's user avatar
  • 347
5 votes
1 answer
117 views

Possible indices of finite index subgroups of $SL_2(\mathbb{Z})$

Here are 3 basic observations regarding $SL_2(\mathbb{Z})$: The abelianization of $SL_2(\mathbb{Z})$ is isomorphic to $\mathbb{Z}/12\mathbb{Z}$ and so $SL_2(\mathbb{Z})$ has a finite index subgroup ...
Hugo Chapdelaine's user avatar
-6 votes
1 answer
87 views

Let $G$ be the special linear group $SL_2(3)$ [closed]

Let $G$ be the special linear group $SL_2(3)$, i.e., the set of all $2 \times 2$ matrices with coefficients from the field $\mathbb{Z}_3$ and determinant equal to 1, with the operation of ordinary ...
math123's user avatar
  • 21
0 votes
1 answer
40 views

Let $a$ be the reflection of the plane $\mathbb{R}^2$ over the bisector of the odd quadrants [closed]

Let $a$ be the reflection of the plane $\mathbb{R}^2$ over the bisector of the odd quadrants (line with equation $y = x$), and let $b$ be the reflection of the same plane over the bisector of the even ...
Markus's user avatar
  • 45
-2 votes
0 answers
48 views

Basis set of a group ring [closed]

Let $(G,\cdot)$ be a finite abelian group and $\mathbb{Q}G=\{\sum {r_g}g:r_g\in \mathbb{Q}, g\in G\}$ be a group ring of $G$ over the rational numbers $\mathbb{Q}$. I would like to know how a basis ...
Kofi Amponsah's user avatar
10 votes
1 answer
165 views

Group of order $n$ is a subgroup of $S_{n-1}$

Let $G$ be a group of order $n$, where $n>1$ is not a prime power. Show that $G$ is a subgroup of $S_{n-1}$. I believe I will need to use some form of Cayley's Theorem and/or Cauchy's Theorem, but ...
Timothy Ho's user avatar
5 votes
1 answer
169 views

Working with character tables

I am currently a bit stumped by an old exam question, which gives a character table and wants me to deduce properties of the group: What is the order of $g$? Show that $g \notin C_G(G)$ Show there ...
Very Interesting's user avatar
1 vote
0 answers
44 views

Normalizer of a Young subgroup in symmetric group

In $S_n$, a Young subgroup $S_{r_1}^{m_1}\times S_{r_2}^{m_2}\times ... S_{r_k}^{m_k}$ where $m_1\times r_1+...+m_k\times r_k=n$ has normalizer $N=S_{r_1}wr S_{m_1} \times ...\times S_{r_k}wr S_{m_k}$....
scsnm's user avatar
  • 1,303
-4 votes
1 answer
85 views

How to prove that all elements inside a cycle of a cyclic group are different from each other [closed]

Let $G$ be a finite cyclic group $(G, \circ)$ and $a \in G$: $$ \langle a \rangle = \{a^z : z \in \mathbb{Z}\}, $$ and $\operatorname{ord}(a) = \min\{a^n : n \in \mathbb{N}_+\}$, i.e., the smallest ...
student129's user avatar
1 vote
0 answers
54 views

IsGroupOfAutomorphisms functionality

I'm looking for two things related to the GAP function IsGroupOfAutomorphisms: whether it does what I think it does (based on the brief GAP manual entry), and if so, how it works. The GAP manual ...
Michael Wynne's user avatar
0 votes
1 answer
57 views

Why is the order of an element equal to the order of the group it generates? [duplicate]

I've found this post Prove the order of an element is the order of the group but this does not help me. Let G be a finite group (G,$\circ$) and a $\in$ G. $ \langle a \rangle $ = {$a^z$ : z $\in \...
student129's user avatar
-1 votes
1 answer
57 views

Can the sum of a nonlinear irreducible character's values on $Z(\chi)$ be zero? [closed]

I need a lemma for a research problem. Suppose that I sum the values of a nonlinear irreducible character $\chi$ of a finite group over the center of that character $Z(\chi)$. Is it possible for the ...
Hanklin's user avatar
  • 15
1 vote
0 answers
42 views

When are subgroups of a 2-Generated group also 2-Generated

In this question it is asked whether subgroups of a finite, 2-generated group are also 2-generated. The answer is no, with a nice counterexample. However, when the group is Abelian, this is true. My ...
Carlyle's user avatar
  • 3,044
1 vote
1 answer
48 views

A basic question about supersolvable quotients

If $G$ is a finite group and $N_i\trianglelefteq G$ ($i=1,2$) such that $G/N_i$ is supersolvable, then, is it true that $G/(N_1\cap N_2)$ is supersolvable? One may give only hint; I will try to prove ...
Maths Rahul's user avatar
  • 3,065
3 votes
1 answer
49 views

Conjugacy classes of normal subgroup in group

Let $G$ be a finite group, and $N$ be a normal subgroup of index $p$. If every conjugacy class of $N$ is also a conjugacy class in $G$, what can we say about $G$ or $N$? Such instances occur if $G$ is ...
Maths Rahul's user avatar
  • 3,065
1 vote
2 answers
138 views

If $G$ has automorphism $a \in Aut(G)$, that only fixes the identity and with $o(a)=2$, then $|G|$ is odd.

Let $G$ be finite group and $a \in Aut(G)$ with $o(a)=2$, if $a(g) \neq g$ for all $g \in G \setminus \{1\}$, then $|G|$ is odd. Hey Guys, I wanted to prove the Theorem above and was wondering if my ...
Stippinator's user avatar
8 votes
3 answers
104 views

Show that $|\rm{Aut}(G)|=48$ for the cube graph

Let $Q$ denote the cube graph (i.e. the graph that looks like a cube in 3D). I am interested in how to prove that $|\rm{Aut}(Q)|=48$. I see that one can write down $48$ elements, so $|\rm{Aut}(Q)|\geq ...
LSt's user avatar
  • 540
0 votes
1 answer
27 views

meaning of $IBr(X | Q)$

In the paper 1, there is a notation used without specifying the meaning. It is $IBr(X | Q)$ in Definition $4.1$. What it means? Irreducible Brauer characters of the group X from a block with defect ...
scsnm's user avatar
  • 1,303
6 votes
1 answer
83 views

Schur’s lemma over $\mathbb{F}_p$

I’m studying modular representation theory, and I got really stuck with the seemingly innocent statement. Consider $\mathrm{GL}_{2}(\mathbb{F}_{p})$ and its center $Z$, which is just a set of all ...
Matthew Willow's user avatar
10 votes
1 answer
364 views

No simple group of order 756 : Burnside's proof

I'm interested in a proof of the non-simplicity of groups of order 756. W.R. Scott, Group Theory, p. 392, exerc. 13.4.9, gives it as an easy exercise, but depending on rather advanced results. I have ...
Panurge's user avatar
  • 1,827
3 votes
1 answer
50 views

Invariants of $2$-torsion group under involution

Let $G=\{1,\tau\}$ be the group with two elements and let $A$ be a free abelian group of finite rank on which $G$ acts (via group homomorphisms). Let $B$ be a $2$-torsion group, also with an action ...
Hans's user avatar
  • 3,615
5 votes
0 answers
37 views

Which transitive $G$-sets appear when repeatedly inducing and restricting $G/H$, where $H\subseteq G$ is an inclusion of finite groups?

Let $G$ be a finite group and $H$ be a subgroup. Then $G/H$ is a transitive $G$-set. We can define a sequence of $G$-sets as follows: $$X_0=G/H$$ $$X_{n+1}=Ind_H^GRes^G_HX_n$$ Like every finite $G$-...
Peter Huston's user avatar
2 votes
0 answers
26 views

Quick question regarding a proof of the 1st Sylow Theorem [duplicate]

Here is the statement and proof: (Note: $N_G(H)$ denotes the normaliser of H in G) If $H \leq G $ and $| H | = p^k$ for some $k < n$, then there is some $P \leq G$ with $H \unlhd P$ and $|P| = p^{...
baslerbuenzli's user avatar
2 votes
4 answers
104 views

Number of non-equivalent irreducible representations of a finite group $G$ over an arbitrary field $F$/non-isomorphic simple $F[G]$-modules

in my algebra class, they give us as an exercise to prove that a finite group $G$ admits at most finitely many non-equivalent irreducible representations over an arbitrary field $F$. Now, I showed ...
F. Salviati's user avatar
3 votes
1 answer
76 views

Exercise 2.19 in Isaacs's book on character theory

Here is the problem: Let $E=\langle x_1,x_2,x_3,x_4\rangle$ be an elementary abelian group of order 16.Let $P=\langle y\rangle$ be cyclic of order 3.$P$ acts on $E$ by $$x_1^y=x_2,x_2^y=x_1x_2,x_3^y=...
Little GTN's user avatar
3 votes
3 answers
100 views

$G=HK$ implies $G = H^xK^y$

"If $H,K \le G$ are subgroups of $G$ and $G=HK$, then $G = H^xK^y$ for all $x,y \in G$." This is an exercise in Isaacs' Finite Group Theory, Problem 1A4. I was able to prove it under the ...
Inbo Gottlieb-Fenves's user avatar
0 votes
1 answer
81 views

Problem 2.14 from Isaacs's Character Theory of Finite Group

I'm solving this problem from Isaacs's Character Theory of Finite Group: Let $H \subseteq G' \cap Z(G)$ be cyclic of order $n$ and let $m$ be the maximum of the orders of the elements of $G/H$. ...
QiQi's user avatar
  • 3
4 votes
1 answer
70 views

Computing all the Galois groups of reducible polynomials of degree 5 over $\mathbb{Q}$

in my Algebra class, it was given as an exercise to find all possible Galois groups of reducible polynomials of degree 5 over $\mathbb{Q}$ without repeated roots. Where, for a field $F$, we call the ...
F. Salviati's user avatar
0 votes
0 answers
60 views

Representations of $\operatorname{PSL}_2 (\Bbb{F}_p)$

Let $p>2$ be prime and consider the group $G=\operatorname{PSL}_2 (\Bbb{F}_p):=\operatorname{SL}_2 (\Bbb{F}_p)/\langle -I \rangle$. Find all complex irreducible representations of $G$. I am aware ...
Robert's user avatar
  • 596
0 votes
0 answers
28 views

Reference for the subgroup structure of PGL2(q)

I have already read the material for the classification of PSL2(q), but I cannot find the full classification of PGL2(q), hence I cannot justify whether I am right or not. If anyone knows where I can ...
Shen Jiahui's user avatar
0 votes
0 answers
34 views

Deduce irreducible representation matrices from the regular representation

I am currently reading chapter 3 from the book Group Theory and Quantum Mechanics by Michael Tinkham. I came across the regular representation $\Gamma^{(reg)}$, more specifically the celebrated ...
Jonasso's user avatar
3 votes
1 answer
59 views

"Almost Retractible" Abelianizations of Groups

I have two related questions. Is there a name for a nonabelian group $G$ whose abelianization is $\bigoplus_{i=1}^n \mathbb{Z}/p_i\mathbb{Z}$ such that for each $i$ there is an element $g$ whose ...
Igor Minevich's user avatar
1 vote
0 answers
60 views

Minimum cardinality of a set on which a perfect group acts transitively

Let $G$ be a finite or numerable perfect group acting transitively on a set $X$. I need to prove that $|X| \geq 5$. I am also working with the extra hypothesis that for all $x \neq y$ in $X$, there ...
Aron's user avatar
  • 263
2 votes
0 answers
20 views

number of lifts of a representation of a finite group to a central extension

Let $G$ be a finite group, and $H\rightarrow G$ a central extension. I'm happy to assume $G$ perfect, or even that it be nonabelian simple. I'd like to understand the relationship between the ...
stupid_question_bot's user avatar
10 votes
1 answer
337 views

Algorithm for finding intersection of two groups from generators

Say I have two subgroups of $S_n$ defined from their generators. E.g. $G_1 = \langle (0 3 4 1), (0 3 2 1 4)\rangle$ and $G_2 = \langle (4)(0 2 3 1), (0 4 3 2 1)\rangle$. Their intersection can be ...
Thomas Ahle's user avatar
  • 4,854
15 votes
0 answers
144 views

Does the sequence $1, 2, 3, 4, 5, 6$ appear in the number of groups of order $n$ up to isomorphism?

Let $\mathrm{gnu}(n)$ denote the number of groups of order $n$ up to isomorphism. $\mathrm{gnu}(1), \mathrm{gnu}(2), \mathrm{gnu}(3), \dots$ is now a sequence of integers, and we may ask if and where ...
Robin's user avatar
  • 3,940
1 vote
1 answer
151 views

Simplest unsolvable quintic with one real root

I am aware that $t^5-t-1$ is unsolvable, but the proof I have seen involves a theorem linking its Galois group with the Galois group of its reduction mod $p$. If I wish to have a simpler proof (that ...
user21820's user avatar
  • 59.2k
1 vote
0 answers
26 views

May we conclude something about a congruence class that $\mathrm{gnu}(2048)$ lies in?

Denote $\mathrm{gnu}(n)$ as the number of groups of order $n$ up to isomorphism. $\mathrm{gnu}(2048)$ is the smallest unknown value of $\mathrm{gnu}$ as of 2024. But, I'm wondering if something of the ...
Robin's user avatar
  • 3,940
-1 votes
1 answer
43 views

map from spin to special orthogonal in Magma [closed]

Let $G:=\operatorname{Spin}(7,5)$. How to construct in Magma the map $G \rightarrow G/Z(G) $ where $Z(G)$ is the center. I get this from Magma: ...
scsnm's user avatar
  • 1,303
1 vote
2 answers
58 views

Determining elementary divisors of a unit group

Currently looking at the following example: Let $G = (\mathbb{Z}/16\mathbb{Z})^\times$. Then $G$ is a multiplicative group with cardinality $\varphi(2^4) = 2^3$. The $\mathbb{Z}$-module-structure is ...
LostInTheSauce's user avatar
0 votes
1 answer
41 views

How to think of regular orbits of a finite orthogonal group?

Let $G$ be a finite subgroup of $O(n)$ acting on $\mathbb R^n$. A regular orbit of the action is one such that the cardinality of the orbit is equal to $|G|$. I am at a loss as to how to prove some ...
rosecabbage's user avatar
  • 1,697
1 vote
0 answers
79 views

Compute the order of group generated by $X \mapsto X+1$ and $X \mapsto\frac{1}{X}$ which are in $\operatorname{Aut}(\mathbb{F}_5 (X) / \mathbb{F}_5 )$

The exercise is as follows: Let $\sigma$, $\tau \in Aut(\mathbb{F}_5 (X) / \mathbb{F}_5 )$, where $\sigma (X) = X+1$ and $\tau (X) = \frac{1}{X}$. ($X$ is the symbol or variable.) Let H be the group ...
shwsq's user avatar
  • 73

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