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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A question about Character Degrees of G/N

Let $G$ be a finite group such that $p_1p_2\mid |G|$, where $p_1$ and $p_2$ are two primes. We know that there exists an irreducible character $\chi\in Irr(G)$ such that $p_1p_2\mid \chi(1)$. We know ...
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286 views

Nonisomorphic groups of order 12.

I'm trying to find 4 groups of order 12, none of which are isomorphic to each other. Should i be trying external direct products? So far i have $A_4, \mathbb Z_{12},\,$ and $\,\mathbb Z_6\times ...
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171 views

Homomorphisms from $D_4$ to $S_3$.

Find all homomorphisms from $D_4$ to $S_3$. We have $D_4 = \{e,r,r^2,r^3,s,sr,sr^2,sr^3\}$ (where $r^4 = e = s^2$) and $S_3 = \{e,(12),(13),(23),(123),(132)\} = \langle (12) (13) \rangle$. Let ...
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1answer
460 views

Group of order $63$ has an element of order $3$, without using Cauchy's or Sylow's theorems

I'm showing that any group of order $63$ has an element of order $3$, and can only use Lagrange's theorem not Cauchy's or Sylow's. I got it reduced to a case of having $62$ elements of order $7$ but ...
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1answer
62 views

A group of odd order has no non-identity elements which are conjugate to their inverse.

I want some verification for my proof to a homework problem. (Is it correct? Is there a simpler way to do this?) Let $G$ be a finite group of odd order and suppose there is an element $g$ that is ...
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625 views

What is the intersection of all Sylow $p$-subgroup's normalizer?

Intersection of all Sylow $p$-subgroups is generally denoted by $O_p(G)$ and it is one of the well studied topics in group theory as there are many theorems related to this. Let $R$ be ...
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101 views

Is the group $G$ cyclic?

Assume that $G$ is a finite group such that for any positive integer $n$ dividing $|G|$, $G$ has one and only one subgroup $H$ with $|H|=n$. Is $G$ cyclic?
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Prove that $N \cap K$ is a normal subgroup in $K$.

Let $N$ and $K$ be subgroup of a group $G$. If $N$ is normal in $G$, prove that $N \cap K$ is a normal subgroup of $K$. Since $N$ is normal in $G$, we have $Ng = gN$ for some $g \in G$. Also ...
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0answers
62 views

Permutation representations

I need a source for Permutation representations of general linear groups over finite fields. Can anyone introduce some sources?
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1answer
46 views

What is the order of this group? [duplicate]

Let $H$ be the subgroup of the group $G$ of all $2 \times 2$ non-singular matrices whose entries are integers modulo a given prime $p$ consisting of those and only those matrices in $G$ whose ...
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2answers
74 views

Representation of $S_4$

Is there a general method to work out all irreducible complex representation of a group? Describe all the the irreducible complex representation of the group $S_4$. $S_4$ is the symmetric group ...
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1answer
356 views

Every finite group of order more than two has a nontrivial automorphism [duplicate]

I want to prove that every finite group $G$ of order more than 2 has a nontrivial automorphism. I've seen this question answered on this site for infinite groups, but the proofs given use the fact ...
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2answers
311 views

How many different elements can we obtain by multiplying all element in a group?

Let $G$ be a finite group. How many different elements can we obtain by multiplying all element in a group? Of course, if $G$ is abelian the answer is one but when G is non-abelian, changing the ...
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1answer
76 views

Nontrivial Homomorphism(s) from $\mathbb{Z_3}$ to $S_3$ - Fraleigh p. 134 13.37

Reference: http://users.humboldt.edu/pgoetz/Homework%20Solutions/Math%20343/hwsome number 1 to 17 that I forgotsolns.pdf There are exactly two nontrivial ...
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1answer
259 views

Sylow p-subgroups, normal subgroups and the center subgroup

Let: $G$ be a finite group. $p$ be a prime number. $P$ be a Sylow-p subgroup of $G$. If $p\mid o(G)$ and for every $(a,b)\in G$, $(ab)^p=a^pb^p$, please help me prove the following: (1) ...
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1answer
376 views

In a finite cyclic group of order n, number of solutions to $x^m = e$ - Fraleigh p. 68 6.53,54

(53.) Show that in a finite cyclic group G of order n, written multiplicatively, the equation $x^m = e$ has exactly m solutions $x$ in G for each $m \in \mathbb{N}$ that divides n. (54.) With ...
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1answer
78 views

Are these two inclusions of finite groups, equivalent?

Definition : Let $\sim$ be the equivalence relation on inclusions of finite groups, generated by : $(H \subset G) \sim (\phi(H) \subset \phi(G))$, with $ \phi: G \to L$ a group morphism, and ...
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How many different group homomorphisms from $\mathbb{Z}/\mathbb{3Z}$ into $SL_{n}(7)$ are there?

I have the following question: How many different group homomorphisms from $\mathbb{Z}/\mathbb{3Z}$ into $SL_{n}(7)$ are there? I think that the answer is the number of elements of order $3$ in ...
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2answers
58 views

How to list all permutations of $S_n$ for any given value of $n$.

In writing out a symmetric subgroup $S_n$ with some given $n$-value, how would I go about doing this? For example, allow me to attempt with $n=3$: $S_3 = \{ (1 3), (2 3), (1 2), (1 2 3), (1 3 2), (2 ...
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Showing $\mathbb{Z}_2 \times \mathbb{Z}_2$ and $\mathbb{Z}_4$ have a different number of subgroups of order $2$.

Looking at this example, can someone explain to me what is $_1$, $H_2$, $H_3$, and how they came about it?
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1answer
60 views

Embedding symmetric groups into general linear groups

Note that any finite abelian group $\Gamma$ can be embedded into $\operatorname{GL}(1,A)$ for some commutative ring $A$. Indeed, let $A=\mathbf Z[\Gamma]$ and $\Gamma\hookrightarrow ...
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1answer
45 views

Conjugate element

Let $G$ be a group of order $2014$. Let $\theta\in G$ such that $|\theta|=19$ and $\alpha\in G$ such that $|\alpha|=2$. Show that $\alpha\theta\alpha=\theta^{\pm1}$. Since the order of $\alpha$ is 2 I ...
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1answer
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general linear groups and definitions

We have two groups, one of them is automorphisms group of a vector space over GF(2) and another one is the direct product of two automorphism group (they are also over GF(2)). Also, via some ...
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Proving that there exists an element of order $p^2$ in a finite abelian group

I've been stuck on this problem for a while now. Let $G$ be finite and abelian. Suppose $\exists x \in G$ such that $x$ has non-square-free order, i.e., $|x| = p^km$ with $p$ prime and ...
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95 views

Estimate the Number of Conjugacy Classes of $G$

This is a series of questions in my book unanswered. Let $c(G)$ be the number of conjugacy classes in $G$. Define $\bar{c}(G):=\frac{c(G)}{|G|}$. Now we estimate the $\bar{c}(G)$ of a non-abelien $G$. ...
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0answers
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Identity based encryption

I am implementing ID based encryption in c# right now i am having problem at the following mathematics $$H_1: \{0,1 \}^n \times \{0,1 \}^n \to \mathbb{Z}^*_p, \text{ what does this expression ...
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1answer
212 views

Character Table of the Monster

Does anyone have a link to a website having the character table of the Griess Monster in characteristic $0$?
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75 views

Class of groups having all subnormal subgroups cyclic

Does there exist a class of non abelian groups whose all subnormal subgroups are cyclic? I searched out by myself but I did not found it.
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1answer
55 views

Abstract Algebra: Index of Subgroups

Here's the problem I'm working on: Prove: Suppose $H$ has index $p$ and $K$ has index $q$, where $p$ and $q$ are distinct primes. Then the index of $H \cap K$ is a multiple of $pq$. (Plus: do you ...
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From linear invariants of group to general ones

There is a lot of information about classical/linear invariants of finite groups. But does it lead to general invariants of group (for example, when we consider some action of our group on finite ...
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1answer
28 views

Understanding semi-direct product construction

I am a student studying semi-direct products for the first time, and have this question:say $G = N \rtimes H$, where $N$ is normal and $H$ is another subgroup that "acts" on $N$. The quotient $G/N ...
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2answers
157 views

Why is studying maximal subgroups useful?

When looking at finite group theory research, it seems to me that a lot of energy is devoted to determining the maximal subgroups of certain classes of groups. For example, the O'Nan Scott theorem ...
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2answers
70 views

Residually finite group with finitely many conjugacy classes of elements of finite order

Let $G$ be a residually finite group. Show that if $G$ has finitely many conjugacy classes of elements of finite order then $G$ has a torsion free finite index subgroup. Not sure how to get started ...
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How many subgroups there is for $\mathbb{Z}_{23}\times \mathbb{Z}_{31}$?

How many subgroups there is for $\mathbb{Z}_{23}\times \mathbb{Z}_{31}$? I try to understand how to solve it but I don't finding a way... I'll be glad if you help me with this... Thank you!
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Calculating sylow subgroups of some concrete groups

Question is to : exhibit all sylow $3$ - subgroups of $S_4$ What i have done so far is : Number of elements in symmetric group is $|S_4|=4.3.2=2^3.3$ number of elements of order $3$ in $S_4$ ...
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1answer
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Finite groups with no common prime factor of their orders

Let $G$ and $H$ be finite groups s.t. their orders have no common prime factor, and let $\phi: G\rightarrow H$ be a homomorphism. I want to show that $\phi(g)=e \space \forall g \in G$ where $e$ is ...
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2answers
371 views

About the converse of Maschke's theorem

The Maschke's theorem say that\ Let $G$ be a finite group and $F$ a field whose characteristic does not divide $\mid G \mid$. Then every $FG$-module is completely reducible (I'm using the notation of ...
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Group theory - Finite or Infinite group

I am just beginning to learn Group theory. As an example of finite groups our Professor wrote this group with just two elements, given by $$ \left( \begin{array}{cc} 0 & z \\ z^{-1} & ...
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1answer
105 views

Semi-direct product in general linear groups

$\operatorname{GL}(n,F)$ can be written as a semidirect product : $\operatorname{GL}(n,F) = \operatorname{SL}(n,F) ⋊ F^\times$ where $F^\times$ is multiplicative group of the field $F$. According to ...
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1answer
61 views

A Problem About Finite Groups

Suppose that $G$ is a finite group and $H,K \leq G$. First prove that $\left | \left \langle H,K \right \rangle :K\right |\geq \left | H:H\cap K \right |$. Then if $\left| H:H\cap K \right |> ...
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1answer
87 views

Classification of automorphism groups of groups of order $p^4$

For the purpose of classifying another algebraic structure which is parametrised by the choice of a group and of an automorphism I need the classification up to isomorphism of automorphism groups of ...
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1answer
130 views

Sylow theorem Cyclic sylow p subgroup

$p$ is the smallest prime dividing order of $G$. $P$ is a sylow p subgroup which is cyclic. Prove that $N_G(P) = C_G(P)$ This is my approach : Since $P$ is sylow p subgroup so its order is some power ...
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225 views

An automorphism on generating set of a group

Let $G$ be a finite group and $A=\{a_{1},...a_{k}\}$ and $B=\{b_{1},...,b_{k}\}$ be two minimal generating sets of $G$ such that $|a_{i}| = |b_{i}|$ for $i=1,\dots,k$. We define ...
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On automorphisms of finite abelian group

Let $G$ be a finite abelian group such that $a, b\in G$ and $\mid a\mid=\mid b\mid$. Then does there exist an automorphism of $G$ such that $\alpha(a)=(b)$? Thank you
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Basic Abstract Algebra - Homomorphism [duplicate]

Given a homomorphism $f:G \rightarrow H$, $G$ finitely generated, what can you say about the order of $g_i$ and $f(g_i)$? I've thought about this question for a while but haven't come to a ...
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1answer
240 views

Groups of order $pq$ have a proper normal subgroup

I am doing the following exercise from [Birkhoff and MacLane, A survey of modern algebra]: Let $G$ be a group of order $pq$ ($p,q$ primes). Show that either $G$ is cyclic or contains an element ...
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1answer
97 views

elementary abelian groups and finite fields

in group theory, an elementary abelian group is a finite abelian group, where every nontrivial element has order $p$, where $p$ is a prime; it is a particular kind of $p$-group. now suppose that we ...
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72 views

Generating Alternating Groups

Is there a way to think about how to generate alternating groups? Say I wanted to generate the alternating groups $A_3,A_4,A_5$.
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108 views

Basic Abstract Algebra - Subgroups of Abelian Group

I'm trying to prove the following: Let $G$ be an abelian group of order 72. Show that $G$ has exactly one subgroup of order 8. I know by theorem that $G$ must have at least one subgroup of order ...
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2answers
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Set of Normal subgroups is a sublattice of a set of subgroups

I need to show that if $ G$ is a group then $\mathcal N(G)$ is a sublattice of $S(G)$. Obviously $N(G) \subseteq S(G) $. How do I show that operations join and meet agree with those of the original ...