A group is an algebraic structure consisting of a set of elements together with an operation that satisfies four conditions: closure, associativity, identity and invertibility. Group theory studies groups.

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What is the notation $PQ$ supposed to mean for subgroups?

My notes just jumps to it without prior explanation like I am supposed to know it. I don't. It's talking about Sylow p-groups and such, $A$ is a group and let $P,Q$ be sylow $p,q$-groups ...
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17 views

Prove that $\frac{\Bbb{Z} \times \Bbb{Z}}{\langle(3,3)\rangle}$ is isomorphic to $\Bbb{Z} \times \Bbb{Z_3}$

I must prove that $\frac{\Bbb{Z} \times \Bbb{Z}}{\langle(3,3)\rangle}$ is isomorphic to $\Bbb{Z} \times \Bbb{Z_3}$. I am trying to do this using the first isomorphism theorem ie. for $\phi: G \to H$ ...
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1answer
60 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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3answers
55 views

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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31 views

Cardinality of a power set? Or is “all subsets of a set” $\neq$ power set?

Again as usual, group theory is muddling me up. A proof of the Sylow $I$ theorem starts as follows Let $X$ be the set of all subsets of $G$ with $|A|=p^m$. where, the setting I have for Sylow ...
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1answer
57 views

How come $(\mathbb{R},\cdot)$ does not form a group? [duplicate]

This is probably a very obvious question but I've only just started learning group theory today. Is the reason that $(\mathbb{R},\cdot)$ does not form a group, because its operation is multiplication? ...
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27 views

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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43 views

$F$ is a field, $n>1$ is it true that $\mathrm{Aut}(F[x]/(x^n))=F^{\times}$?

My question is to ask for whether the following generalization of this problem is true or not. The mentioned problem says $A:=\mathrm{Aut}(\mathbb{Q}[X]/(X^2)) \simeq \mathbb Q^{\times}$ by proving ...
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21 views

$H,K$ be subgroups of a group $G$ such that $HK$ is also a subgroup ; when is it possible that $HK$ is a homomorphic image of $H$?

Let $H,K$ be subgroups of a group $G$ such that $HK$ is also a subgroup ; when is it possible that there exist a surjective group homomorphism from $H$ onto $HK$ ? If both $H,K$ are finite then it is ...
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1answer
24 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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Can any presentation of a finitely presented group be reduced to a finite one?

Suppose $G = \langle x_1, \ldots, x_n \mid p_1, \ldots, p_m \rangle$ is a finitely presented group, and let $\langle A \mid R \rangle$ be another presentation of $G$, with $A$ and $R$ possibly ...
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3answers
34 views

What is the logic in Free groups here

Free groups have been hurting my brain for months and this one in particular, I cannot comprehend what is going on with logic $H= \langle a,b\mid a^{-1}ba=b^2,b^{-1}ab=a^2\rangle$. Show that $H$ ...
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2answers
14 views

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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1answer
21 views

The bit from Armstrong's book on Group Theory; Theorem 10.2

The below is the theorem concerned, and the bit in red I have circled I cannot understand. Issue is, it says "if." Well yes, "if" it is true hen indeed $xy=x'y'$. But has it been proved that that is ...
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1answer
21 views

Finding a representative cocycle for a given group extension

Suppose we have a group extension $$0 \to N \stackrel{\iota}{\to} E \stackrel{\pi}{\to} G \to 1$$ where $N$ is abelian. How to find a representative 2-cocycle that produces this extension? Or more ...
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5answers
36 views

Prove $G$ is Abelian if $N$ is in the centre of $G$ and $G/N$ is cyclic

I need some help on this one. $G$ is a group. If $N$ is a subgroup of $G$ contained in the centre of $G$ and $G/N$ is cyclic, show that $G$ is Abelian. My attempt is only half way and stuck at ...
2
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1answer
51 views

bounded generation and groups with infinitely many ends

Following section 7.1 in Peterson-Thom's paper here, we say a countable group $G$ is boundedly generated by the subgroups $G_1, \cdots, G_n$, if there exists an integer $k\in\mathbb{N}$, such that ...
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1answer
41 views

Finding all Group Homomorphisms

Take $\mathbb{Z}_{15}$ and $\mathbb{Z}_{18}$, and find all the group homomorphisms. I'm having trouble with the concept, so the specific example is not necessarily important. From the definition of ...
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2answers
33 views

Non-abelian, simple subgroups of $S_n$

I am trying to prove, as part of a larger theorem, that if $G$ is a non-abelian finite, simple group of order $>2$ and $G$ is a subgroup of $S_n$, then $G$ must be a subgroup of $A_n$. Any ideas ...
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2answers
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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45 views

Prove that G is an Abelian Group

Let G = {x in Q : 0≤x<1}. Define the operation on G: a•b = a+b if 0≤a+b<1, a+b-1 if a+b≥1 Prove that (G,*) is an Abelian group. Attempt: (commutativity was easy). For associativity I got a+b≥1 ...
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18 views

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
10 views

Group-theory,discrete mathematics, trichotomous property

Is the intersection of trichotomous relations trichotomous? Generally, trichotomy is the property of an order relation < on a set X that for any$ x$ and $y$, exactly one of the following holds: ...
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36 views

How to determine whether or not $S_{n}$ contains a subgroup of order $m$

I'm currently going through group theory practice problems and I needed some assistance with the following exercise: Does $S_{9}$ have a subgroup of order $25$? Is it correct to just use Lagrange's ...
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2answers
30 views

How can I compute these values in groups? [closed]

In the group of $\mathbb{Z}_{17}^* $, $ \overline{13}^{-1}=?$
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1answer
42 views

Prove that $G=\langle g_1,…,g_n\rangle$ if $G$ abelian.

Let $G$ a finite abelian group. Are there $g_1,...,g_m$ s.t. $$G=\langle g_1,...,g_m\rangle$$ where $g_i\neq g_j^m$ for all $m$ and all $i\neq j$ ? For cyclic group no problem, but I can't see such a ...
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45 views

A Group G is abelian $\Leftrightarrow$ $ Inn(G)$ is a normal subgroup of Sym(G)

First of all I don´t think that this question is answered here If $G$ is non-abelian, then $Inn(G)$ is not a normal subgroup of the group of all bijective mappings $G \to G$ because in my opinion ...
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19 views

Find an isomorphism $PGL_2(F_3) \cong S_4$

I am struggling to find an explanation why this true, I know and I'm sorry that kind of question is commonly asked , although I couldn't find anything about this particular question. Help is ...
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24 views

Criterion for a group $G$ to be abelian using $Inn(G)$ [duplicate]

Let $(G, *)$ be a group. Let $\mathcal S(G)=\{f:G\to G\space|f\space is\space bijective\}$ be the symmetric group and $Inn(G)=\{\kappa_{a}\space|\space a\in G\}$ the inner automorphisms. Now I want to ...
2
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1answer
36 views

example of a group which has no proper subgroup of finite index, but it does have maximal subgroups.

Let G be a group. if G has no proper subgroup of finite index, can we say that it has no maximal subgroup? if it is not true, what's the counterexample for this assertion?
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1answer
21 views

Subgroup question

From Jacobson's Basic Algebra I , exercise 9 of section 1.2: Let $G$ be a non-vacuous subset of a monoid M. Show that G is a subgroup if and only if every $g \in G$ is invertible in M and ...
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2answers
29 views

Show that $\left|PGL(2,q)\right| = (q-1) q (q+1)$

I have thought about quite a while, I figured that if $q = 2$ or $q = 3$, then $(q + 1)! = (q + 1)(q)(q − 1)$ because $0! = 1$ and $1! = 1$ respectively. I'm sure that I can use this somehow but I ...
0
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1answer
43 views

determine whether group is additive

I am taking a stab at group theory and in some of the questions I am working on they don't explicitly state whether a group is additive or multiplicative. $\mathbb{Z}_4$ is additive (and that makes ...
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0answers
29 views

Is G isomorphic to a Subgroup of $GL(2,\mathbb C)$

I'm stuck on a question on a past exam paper that asks if a group $G$ is isomorphic to a subgroup $GL(2,\mathbb C)$. We are given the character table for $G$ which I've attached below. It's the last ...
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40 views

Units in a group ring $\mathbb{Z}[G]$

This is a homework question: For a group G, let $g\in G$ have finite order, such that $\langle g\rangle$ is not a normal subgroup of $G$. Then $\mathbb Z[G]$ has a unit other than $\pm h$ with ...
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3answers
85 views

An exercise to understand free group

I am new to the concept of free groups, reading Artin's Algebra but completely lost. So I hope I can learn from concrete examples instead of theorems corollaries. So I jumped to the exercise, and here ...
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3answers
52 views

Constructing a surjective morphism $D_4 \to \Bbb Z_2$

I'm given two groups: $D_{4, \circ}$, which is the dihedral group of order $8$, consisting of the elements $1, a, a^2, a^3, b, ab, a^2b, a^3b$; here, $a$ means a rotation and $b$ a reflection the ...
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0answers
34 views

Trivial intersection of two normal subgroups

Suppose that $M$ and $N$ are two normal subgroups of a group $G$ such that $M\cap N = \{1_G\}$. Let $A, B$ are subgroups of $M$. If $AN=BN$ then $A=B$. I start by showing $A\subseteq B$. Let ...
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1answer
94 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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1answer
18 views

Characteristic of an integral domain is 0 (or prime) [duplicate]

The characteristic of an integral domain $R$ is $0$ (or prime). My lecture has not yet covered infinite integral domain but I'll like to understand the proof. Basic fact: $R$ is an ...
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1answer
25 views

If $H$ is a subgroup of $G$ of index $2$ and $K$ is a subgroup of $G$ not contained in $H$, then $HK = G$

Let $H$ and $K$ be two subgroups of a group $G$ such that $[G : H]=2$ and $K$ is not a subgroup of $H$. Then show that $HK=G$. Now, since $HK$ is a subset of $G$ we need only to show that $G$ is a ...
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2answers
38 views

If $G$ is a cyclic group with exactly two nontrivial proper subgroups, prove $|G| = pq$ or $|G| = p^3$

I can prove that $G$ is cyclic, but I am not sure how to prove the orders. I know I need to use the Fundamental Theorem of Cyclic Groups but I'm not sure how to apply it. Is there something obvious I ...
3
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1answer
69 views

Every finite group is isomorphic to the Galois group of some polynomial

I was reading through chapter 14 of Dummit and Foote just now and I came across the sentence "It is an open problem to determine whether every finite group appears as the Galois group for some ...
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0answers
30 views

Composition Series Analogous to Compactness?

The wikipedia page for group with operators makes the following claim about composition series as being analogous to compactness: The Jordan–Hölder theorem also holds in the context of operator ...
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28 views

Holomorph of Cyclic $p$-groups

Assume that $G$ is a cyclic $p$-group (or up to isomorphism more explicitly $\mathbb{Z}_{p^r}$). Do you know if there does exist any explicit way to describe the holomorph of the above group? ...
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1answer
32 views

Order of subgroup of symmetric group

Let $X$ be a finite set, i.e. that $|X| = n$, and let $G = \operatorname{Sym}(X)$ be the symmetric group on $X$. Let $Y \subseteq X$ be a subset of $X$ and define the subset $G_Y \subseteq G$ to be ...
3
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1answer
70 views

groups with infinitely many ends are not boundedly generated?

Recall that a group $G$ is boundedly generated if it can be written as a finite product of cyclic subgroups. And there are a lot of examples of groups that are (not) boundedly generated. I am ...
2
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1answer
29 views

Find the largest positive integer $k$ such that $S_5 \times S_5$ has an element of order $k$.

Find the largest positive integer $k$ such that $S_5 \times S_5$ has an element of order $k$. I know by Lagrange that the order of any element of $S_5\times S_5$ divides $\left\lvert S_5\times ...
3
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3answers
40 views

Show that $H \cap K$ is an abelian subgroup

Let $G$ be a finite, multiplicatively written group and let $H$ be a subgroup of $G$ of order $33$ and let $K$ be a subgroup of $G$ of order $77$. Show that $H ∩ K$ is an abelian subgroup of $G$. I ...
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0answers
14 views

Sum over trace of $U(n)$ infinitesimal group generator product?

Consider the group $U(n)$ and a set of $n^2$ generators $T_a,~a=1,2,...,n^2$ of its infinitesimal algebra. Consider taking a trace of a product of some amount of these generators: ...