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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Character theory - exercise 3.4 from Isaacs

Let $G$ be a simple group and suppose $\chi\in Irr(G)$ with $\chi(1)=p$ a prime, Show that a Sylow $p-$subgroup of $G$ has order $p$. The indication provided is: if the Sylow $p-$subgroup $P$ is ...
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Prove that $\alpha$ is an automorphism of $Z_n$.

Let $ r \in U(n)$[We define $U(n)$ to be the set of all positive integers less than $n$ and relatively prime to n]. Prove that the mapping $ \alpha : Z_n \rightarrow Z_n $ defined by $ \alpha(s)=sr$ ...
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21 views

Show that a set is a right transversal for $S_3$ in $S_4$

I am given the set A ={e,(14),(24),(34)} I'm supposed to show that this set is a right transversal for S3 in S4, meaning that every right coset of S3 contains exactly one element of A. I'm getting ...
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15 views

Number of conjugacy classes or bounds for the index of stabilizers on finite Coxeter groups.

For a finite group $G$ denote by $\operatorname{conj}\left(G\right)$ the number of conjugacy classes of $G$ and let $q\left(G\right)$ be the quotient: \begin{align} q\left(G\right)=\frac{\...
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60 views

What is the smallest number $n$ with $gnu(n)=2016$?

$gnu(n)$ denotes the number of groups of order $n$ upto isomorphy. What is the smallest number $n$ with $gnu(n)=2016$ ? The numbers upto $n=2047$ do not satisfy the given property. I do not know ...
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1answer
50 views

$p$-groups have normal subgroups of each order [duplicate]

Suppose $|G|=p^n$. Then $G$ has a normal subgroup of order $p^m$ for every $0\le m\le n$. By induction. It is clearly true for $n=0$. Now suppose $k<n$ and $H_i$ is a normal subgroup of $G$ of ...
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Explicit computation of $H^2(\mathbb{F}_p^n, \mathbb{R}/\mathbb{Z})$.

I'm interested in the computation of the second cohomology group of the elementary abelian group $\mathbb{F}_p^n$ with coefficients in $\mathbb{R}/\mathbb{Z}$: $$H^2(\mathbb{F}_p^n, \mathbb{R}/\...
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1answer
51 views

Determining the number of subgroups of $\Bbb Z_{14} \oplus \Bbb Z_{6}$

I want to determine how many subgroups does the additive group $G:=\Bbb Z_{14} \oplus \Bbb Z_{6}$ have? There are many related posts in our site, for instances: here and there. However, it ...
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64 views

Chief factors and local formation

Every thing below is concerned with finite groups. My question is about this paper A class of groups is a collection $\mathcal{X}$ of groups with the property that if $G \in \mathcal{X}$ and if $H \...
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77 views

Show that $S_5$ does not have a quotient group isomorphic to $S_4$

Show that $S_5$ does not have a quotient group isomorphic to $S_4$. If we to assume that $H$ is such a group, than $H$ must be normal in $S_5$ and $|H|=|S_5|/|S_4|=5$. So $H$ must be isomorphic to $\...
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49 views

Question about finite abelian group

Let G be an abelian group of order $mn$ where $\gcd(m,n)=1$. I proved that $mG$ and $nG$ are subgroups and that $G=mG+nG$ and now i want to prove the three things: the sum is direct, i.e. $mG\cap ...
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1answer
31 views

About the conjugacy classes of a finite group

Let $K_1, \cdots , K_n$ denote the conjugacy classes of a finite group $G$. For $x \in K_s$, define $n_{ijs} = |\{(y, z) \in K_i × K_j : yz = x\}|$. I want to show that $n_{ijs} = n_{jis}$. How ...
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32 views

Automorphism Group of a cyclic p-group

I want to show that the automorphism group of $C_p^{k}$ is cyclic for an odd prime $p$. I know that the order of $Aut(C_n)$ is $\phi(n)$ and so the order of $C_{p^{k}}$ is $\phi(p^{k}) = p^{k-1}(p-1)...
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Show that all elements of $\left<a,b,c\right>$ are of the form $a^ib^jc^k$ (comprehension)

Let $G=\left<a,b,c\right>$ a subgroup of $\mathfrak S_6$ where $a=(123),b=(456)$ and $c=(23)(45)$. Show that every element of $G$ can be uniquely written as $a^ib^jc-k$ where $0\leq i,j\leq 2$ ...
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Looking at two versions of Fundamental Abelian groups theorem

$G$ is a finite abelian group. Then it can be expressed with a direct product of cylic groups with prime power order $G$ is a finite abelian group with order $n$. Then it can be expressed as ...
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135 views

Do there exist simple groups of order $n(n+1)$ for some integer $n>1$?

In the comments to this question, it was remarked that the question of whether a finite group of order $p(p+1)$ is necessarily not simple becomes fairly interesting if we do not assume that $p$ is a ...
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1answer
31 views

Why does Lagrange come into play here? [duplicate]

I am given this problem, and the solution reasons differently from me and obscurely, to me Let $G$ have order $30$. Show that it is not simple. The solution is apparently So, what I don't ...
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32 views

Sylow theorem; Sylow $III$

Just curious, I have't seen any statement that answers this so far. And I'm not at the stage of being able to or given a proof of the theorem so I am not sure. Sylow $III$; If $G$ has order $n=p^...
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33 views

Are the normalizers of Sylow p-subgroups isomorphic?

Let $G$ be a finite group and $A,B \in \text{Syl}_p G$, for some prime $p$. Is it always true that the normalizers $N_G(A)\cong N_G(B)$? I just need a hint to get started, because I don't know where ...
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1answer
90 views

Group of order $p^2+p $ is not simple [duplicate]

Can some one please give me a hint to prove that every group of order $p^2+p$ is not simple?
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1answer
22 views

first homology group with coefficients in divisible group

I had (perhaps very elementary) doubt in the understanding of the computation of first homology group of a finite group over a divisible group. Let $\pi$ be a finite group of order $n$ and $D$ be a ...
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24 views

The point of a group-theoretic Chinese Remainder Theorem?

It states that for coprime $m,n$ nonzero integers, $C_{mn} \cong C_m \times C_n$. However, I know a theorem that says Cyclic groups with the same order are isomorphic. So $C_{mn} \cong C_m \...
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36 views

Non-abelian groups of order $50$

Are there non-abelian groups of order $50$? If so, how many elements of order $2$ can they have? Such a group would have a unique $5$-Sylow-group, so it would have at least $25$ elements not of order $...
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1answer
19 views

Crossed homomorphism from semi-direct product: confusion in definition

(Ref: this) Let $\pi \times_{\varphi} G$ be semi-direct product in which $G$ is normal and $\pi$ is complement. Let $\omega$ be another complement of $G$ in above semi-direct product (so $\pi \...
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25 views

Ordering of elements in the base of a group

In section 4.6.7 of HANDBOOK OF COMPUTATIONAL GROUP THEORY, the authors use an ordering $\prec$ for the elements in a coset. That ordering, $\prec$, was defined in section 4.6 as follows. Throughout ...
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1answer
27 views

Website or book with Hasse diagrams of subgroups

I need to look at Hasse diagrams of very many groups, especially high powers of small symmetric groups. Is there any place where I could look them up? Calculating them myself would be a huge amount of ...
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1answer
61 views

What does being Abelian have to do at all with the proof?

I don't understand why the proof needs to consider cases that $G$ is Abelian and non-Abelian. If $|G|=p^n$ where $n>1$ then show that $G$ cannot be simple. It uses the theorem If $G$ is a ...
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Abelian finite group [duplicate]

This is a (maybe be simple) problem from Group Theory, but being a beginner, I am unable to take even a first step forward. Let $G$ be a finite group whose order is not divisible by $3$.Suppose that $...
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Does “order of a subgroup $n$” mean “there is an element of order $n$ in $G$”?

I am not sure about this. I understand it when it is cyclic. But if not stated so, I cannot reason as to why.($G$ here I assume finite) $G$ is a group with some subgroup $H$. Then, if $|H|=n$ then ...
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1answer
30 views

Permutation and Linear Representation of Finite Group

By a permutation representation of a finite group $G$, we mean a homomorphism from $G$ to $S_n$, the (full) permutation group on $n$ letters. By a linear representation of a finite group $G$, we mean ...
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1answer
46 views

Representation of Sylow which does not extend

Let $H$ be a subgroup of a finite group $G$ and $\rho$ a representation of $G$ such that the restriction of $\rho$ to $H$ is invariant under conjugation in $G$, in the sense that its character is ...
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Showing that the primary component $G_p$ is a subgroup of $G$

For a finite abelian group $G$ and a prime number $p$ with $p \mid |G|$, we define $G_p$ as the subset of $G$ that contains all elements of $G$ with order $p^k$ for a $k \in \mathbb{N}_0$. We call $...
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48 views

Show that $x^{n-1} =1$ for all non-zero element in a field

Question: Show that $x^{n-1} =1$ for all non-zero element in a field Let F be a finite field of order n. Show that $x^{n-1}=1$ for all non-zero $x \in F$. We have $\left | F \right |=n.$ ...
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Groups and queues and stacks

As I review my elementary CS material to prepare for an interview I cannot help but think that I missed a key connection when studying this prior: I think I missed the relationship between operations ...
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1answer
96 views

About subsets of finite groups with $A^{-1}A=G$ or $AA^{-1}=G$?

Regarding to the problems Does $A^{-1}A=G$ imply that $AA^{-1}=G$? and Is it true that if $|A|>\frac{|G|}{2}$ then $A^{-1}A=AA^{-1}=G$?, we are looking for some subsets $A$ of $G$ with $\lfloor \...
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Show that the permutation $(1 \space 2 \space 3)$ can not be a cube of any element of $S_n.$

Here is my try: If there exists $a \in S_n$ such that $a^3=(1 \space 2 \space 3)$, then $a^9=e$ where $e$ is identity in $S_n$. Then $o(a)=9$. I don't know how to proceed further. Can anyone ...
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1answer
38 views

Study of specific Quotients of a $p$-group in MAGMA

In Magma, I wanted to do a program, but since I was getting errors in "basic" commands, I am in trouble. Here I will describe a problem to be done in MAGMA, then its interpretation in MAGMA. I would ...
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17 views

Primary and Secondary invariants for finite groups

For a finite group G and complex representation V of degree n, I would like to know the precise definition of Primary invariants. Does any set of n algebraically independent homogeneous invariants ...
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1answer
25 views

$p$-Groups and the Cauchy theorem

Here is the definition of a $p$-group $p$ is prime. A $p$-group is a group $G$ such that every element has an order of a power of $p$. So let me check my understanding, every element of $G$ has ...
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Conjugacy Class Equation for $\mathbb{Z_{25}}$

I'm suposed to find the conjugacy class equation for $\mathbb{Z_{25}}$. Since $\mathbb{Z_{25}}$ is Abelian, that means that $gxg^{-1}=xgg^{-1}=x$ so $Z(\mathbb{Z_{25}})=\mathbb{Z_{25}}$ and every ...
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Kazhdan-Lusztig polynomials for the longest element in finite Coxeter groups

Suppose $W$ is a finite Coxeter group and that $w_\circ$ is the longest element. I want to prove that $P_{x,w_\circ}=1$ for all $x \in W$. I know from Corollary 7.14 in the book Humphreys p. 167 ...
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1answer
28 views

Relation between conjugate subgroups and the subgroup generated by them

Let $G$ be a group with $H\leq G$. Suppose that $g\in G$ and $y\in \langle H, gHg^{-1}\rangle$. If $yHy^{-1}$ and $gHg^{-1}$ are conjugate in $\langle yHy^{-1}, gHg^{-1}\rangle$, then $H$ and $gHg^{-...
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What is the name of this finite group of order 36

In my research we find a finite group of order 36, which satisfies the following generating relations \begin{equation} \langle g_{12}~,~g_3 ~|~g_{12}^{12}=e=g_3^3, ~~~g_{12}~g_3=g_3^2~g_{12}\rangle \...
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3answers
60 views

Group of order $p^2$ [duplicate]

Let $G$ be a group of order $p^2$ where $p$ is a prime. I want to show that either $G$ is cyclic or is the product of two cyclic groups of order $p$. After some work, the problem reduces to showing ...
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1answer
27 views

Showing a translation group is a normal subgroup of an affine group

Let V be an n-dim vector space over the field $\mathbb{F}.$ $A \in GL\left ( n,\mathbb{F} \right )$ and $v \in V$ Define the affine transformation $t_{A,v}$: $V\rightarrow V$ $x \mapsto xA+v$ ...
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44 views

Proof of the fixed point theorem in group action.

$\textbf{Fixed point theorem:}$ Let $G$ be a $p$-group and let $S$ be a finite set on which $G$ operates. If the order of $S$ is not divisible by $p$, there is a fixed point for the operation of $G$ ...
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34 views

How does the Whitehead quadratic functor act on morphisms?

I am trying to understand the universal quadratic functor $\Gamma$ that appears on the Whitehead exact sequence (J. Whitehead, A certain exact sequence, Annals of Mathematics 52 (1950)) particularly ...
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19 views

Showing the affine transformation is well-defined

Let V be an n-dim vector space over the field $\mathbb{F}.$ $A \in GL\left ( n,\mathbb{F} \right )$ and $v \in V$ Define the affine transformation $t_{A,v}$: $V\rightarrow V$ $x \mapsto xA+v$ ...
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30 views

PGL(2,c) is isomorphic to PSL(2,c)

Definition: Projective special linear group $PSL\left ( n,\mathbb{F} \right )=\frac{SL\left ( n,\mathbb{F} \right )} {\left ( Z\left ( GL\left ( n,\mathbb{F} \right ) \right )\cap SL\left ( n,\...
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1answer
23 views

Language used in projective linear group

In lectures and text on topic of projective linear group, I hear and see the word "factor out" or "quotient out" thrown around a lot. What is the word supposed to mean? If this is vague, I can ...