Abstract algebra is the study of algebraic objects. Some of the more common algebraic objects are groups, rings, fields, vector spaces, modules, and other advanced topics.

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The question on a ring.

let $$G=\{5m+7n | m, n\in \Bbb N\}$$ Firstly, I want to find the complement of $G$ In $\Bbb N $ is finite. Secondly, how do I find the Frobenius number of $G$ (I guess, the larger not ...
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$\exists a\in G-H$ such that $aHa^{-1}=H$

Let $G$ be a $p$-group with proper subgroup $H$. Show that there exists an element $a\in G -H$ such that $a^{-1} Ha = H$ Can you check my proof? Since $G$ and $H$ are $p$-groups their centers ...
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1answer
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Notation question: Group generated by two elements?

Let there be $H$ subgroup of symmetric group $S_4$, so that $H=<(12)(34),(234)>$. What does the notation $<(12)(34),(234)>$ mean? I know that if there's one elements, then it's all the ...
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2answers
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Sylow p-subgroups: Understanding a proof

I don't understand the last part of this proof: http://www.proofwiki.org/wiki/Intersection_of_Normal_Subgroup_with_Sylow_P-Subgroup where they say: $p \nmid \left[{N : P \cap N}\right]$, thus, $P ...
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Show that $c-b\lt b-a$

The question: Let $G$ be an Arf semigroup and $a\lt b\lt c$ be three consecutive elements in $G$. How to show that $c-b\lt b-a$ and how to show that this is not necessarily the case for every ...
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3answers
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Show that any finite extension of $\mathbb{Q}$ is not algebraically closed.

EDIT: Entirely wrong question. I wanted to ask something else. How do I show that any finite extension of $\mathbb{Q}$ is not algebraically closed. In other words, the algebraic closure of ...
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1answer
22 views

How to prove this binary operation? [duplicate]

Let (G, ∗) be a group and a,b € G. Show that (a✻b)^-1 = a^-1✻b^-1 if and only if a✻b=b✻a I could not solve this, how can ve prove it? a^-1 means inverse of a
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24 views

How to prove this binary operations

Let $(G, ∗)$ be a group and $a,b \in G$. Show that if $(a∗b)∗(a∗b) = (a∗a)∗(b∗b)$, then $(a∗b)=(b∗a)$. How can ve prove this? Thanks for your help..
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1answer
21 views

$Z$ is cyclic and has generators 1 and -1

I know that group G is cyclic if there exist $ g \in G$ such that $ G = \{g^k : k \in Z\}$. However I don't understand how Z has generators $1$ and $-1$. Does $g^k$, so in this case, $1^k$ mean ...
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1answer
20 views

Finding explicit form of group homomorphism

Let there be $f: \mathbb Z_{50} \rightarrow \mathbb Z_{15}$ group homomorphism so that $f(7)=6$. Find explicit form of $f$. What's the approach to this type of questions? Is it possible that the ...
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primes splitting completely in cyclic extensions

Let $K$ be a quadratic number field. It is a well known result that a prime $p$ splits completely in $K$ if and only if $\left(\frac{d_K}{p}\right)=1$. What about cubic extensions? Can we find ...
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Field-like Algebraic structure with infinite additive identities

Suppose I have a field-like structure with a set $F$ and two operations (addition and multiplication) and their respective inverses. It respects the following proprieties similar to a field: ...
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2answers
30 views

Group having an element $x$ of order $p$ where $p$ is the smallest prime dividing |G|

Let $G$ be a finite group and $p$ be the smallest prime dividing $|G|$ and $x\in G$ be an element of order $p$. Let $h\in G,$ and $hxh^{-1}=x^{10}$. Then prove that $p=3$. If $H=<h>, ...
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11 views

Part of a proof in Herstein about Gaussian Integers being a Euclidean Ring

In Herstein topics in algebra (2nd Edition) page 150, in proof of theorem 3.8.1, in the first special case where $n$ is a positive integer and $y=a+bi$ is a general Gaussian integer, where he is ...
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1answer
11 views

Fields with Additive identity powers

Would it be possible to have a field (or field-like structure) with an additive identity $k$ where $k^a\neq k^b$ for $a\neq b$? I need this because I'm working with a field-like structure where if I ...
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2answers
49 views

Is there any motivation for the direct sum?

I believe that everything have reason to exist . I want to learn why the direct sum exist. Is there any motivation for the direct sum to exist ?
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Abstract algebra notation

I am new to learning abstract algebra and using multiple books but the notation varies enough to throw me off. Could someone explain to me the differences between: $\mathbb{Z}\left\{ p\right\}$ ...
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1answer
15 views

Question about proof about index and subgroups

Let $G$ be a group so that $H\lhd G$. There is an element $g \in G$ so that $g$ isn't in $H$ but $g^2$ is in $H$. Show the index is even. Can't I just say that the cosets of $H$ are $H$ and $Hg$ ...
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3answers
22 views

A question about normal subgroups and index

Let $G$ be a group, and $H$ be a normal subgroup of $G$. $|H|=11$ and $[G:H]=24$. Let there be $x \in G$ and $x^{11}=e$. Show $x \in H$. Would like hints etc' on how to solve this. Is proving that ...
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1answer
21 views

Groups with single conjugacy class of subgroups. (modified)

I am going to modify my previous question. What are those finite non abelian groups in which non normal subgroups of same order are conjugate. e.g. Dihedral groups of order $4n+2$.
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Why is the reciprocal of an $n$-th root of unity its complex conjugate?

As stated in the Wikipedia article on roots of unity, the reciprocal of an $n$-th root of unity is its complex conjugate. They provide the following proof of this statement: Let $z\in\mathbb{C}$ be a ...
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6answers
53 views

Why is $\mathbb{R}/\mathbb{Z}$ isomorphic to the complex numbers of length one?

Wikipedia states that the quotient group $\mathbb{R}/\mathbb{Z}$ is isomorphic to all complex numbers of length $1$. I have a hard time making sense out of this, and in particular, how complex numbers ...
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31 views

Let $G$ be a group. Then verify the statements with justification:

Let $G$ be a group. Then verify the statements with justification: $\bullet$ If $G$ has nontrivial centre $C$, then $G/C$ has trivial centre. $\bullet$ If $G$ does not equal $1$, there exists a ...
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Describing $k$-automorphisms

(1) Let $B=k[T]$ with $k$ a field. Describe the group $Aut_kB$ of $k$-automorphisms of B. (2) Do the same for the field $K=k(T)$ or rational functions.
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38 views

Question about group theory and order in $\mathbb Z_n$

This is a only theoritical. Why is the order $o( \bar x )$ of $\bar x∈\mathbb Z_n$ the smallest non-negative integer $k$ such that $kx \equiv 0$ (mod $n$)? I don't understand how it follows from ...
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13 views

How to show that a cyclic group is partitioned into these classes under a equivalence relation.

Condition: $S=$ {$h^k|k\in \mathbb Z$}, $h^{k_1}\sim h^{k_2}$ iff $k_1 \equiv k_2$ $mod$ $n$, Then, how to show that $S=[h^0]\bigcap[h^1]\bigcap[h^3]\bigcap...\bigcap[h^{n-1}]$?
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3answers
52 views

Only $1$ Nontrivial Subgroup $\Longrightarrow |G| = p^2$ [duplicate]

I am pretty new to this site , so I am not sure how things work, but I am in desperate help with a question that I don't know where to start or finish with. It is for a test I have to study for. Here ...
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1answer
34 views

Can we give of the fact that a group of order $9$ is abelian without using an argument involving the product of two cyclic groups of order $3$?

A group of order $9$ is always abelian. I've seen proofs of this result, but I would like to prove it the following way: Let $G$ be a group of order $9$. If $G$ has an element $a$ of order $9$, then ...
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26 views

Show that $(3,1)$, $(-2,-1)$, and $(4,3)$ generate the additive group $\mathbb{Z} \times \mathbb{Z}$

Show that $(3,1)$, $(-2,-1)$, and $(4,3)$ generate the additive group $\mathbb{Z} \times \mathbb{Z}$. I need your help.
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1answer
33 views

Visualizing the 48 actions of GL(2,3)

Hello and thank you for your patience. (DISCLAIMER: I'm a novice and not a mathematician by trade and I'm not certain how to articulate most of my questions here. I am learning from experiences and ...
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1answer
23 views

Generators of the additive cyclic group $\mathbb{Z}$

Show that the only generators of the additive cyclic group $\mathbb{Z}$ are $1$ and $-1$. I need your help.
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1answer
48 views

What sort of algebra is this?

Let us say that I have a set of symbols, $S$. The symbols can be operated on by a set of $n$-ary operators, $O$. Importantly, some of these operators are in the set of symbols, i.e. $S \cap O \neq ...
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2answers
52 views

What is the most elementary proof of the following: a non-abelian group of order $6$ is isomorphic to $S_3$

We know that, a non-abelian group of order $6$ is isomorphic to $S_3$. While I've been able to locate different proofs of this result, I would like to have one that is as elementary as it can be. So ...
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1answer
29 views

An assumption used in defining a group of order $mn$

I want to show that the defining relations $a^m=b^n=e, ab=ba^k$ define a group of order $mn$ with a normal subgroup of order $m$, if $k^n \equiv 1 \pmod m$. Consider the set of symbols of the form ...
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35 views

How do I effectively solve this kinds of problems?

I'm preparing for a test on Monday. This is an exercise in the past homework. Find all subgroups of $\mathbb{Z}_2\times \mathbb{Z}_2\times\mathbb{Z}_4$ isomorphic to $Z_2\times Z_2$. Is there a ...
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1answer
22 views

Lie bracket of vector fields on $R^2$

Compute the Lie bracket$$\Big[-y\frac{\partial}{\partial x}+x\frac{\partial}{\partial y},\frac{\partial}{\partial x}\Big]$$ on $R^2$ Can you help me please?
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1answer
27 views

The first Weyl algebra is Calabi-Yau

Why is the Weyl algebra $A_1(k)$ over a field $k$ Calabi-Yau? (My definition of Calabi-Yau is Ginzburg's)
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1answer
21 views

Non simplicity of a group of order $p^{100}q$ given some conditions.

Let $G$ be a group $p$, $q$ primes $p\neq q$, $|G|=p^{100}q$. 1.- Assume that $G$ has two different $p$-Sylow subgroups and intersection for each pair of $p$-Sylow subgroups is trivial. Show that ...
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1answer
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Why the rank of finitely generated finite abelian group is 0?

I haven't proven 'the fundamental theorem of finitely generated abelian group' Nevertheless, it's written in my text(Dummit,Foote - p.159) it's gonna be proven in a later chapter. Also, it's written ...
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2answers
31 views

$\mathbb{Z}/(p)^n$ does not contain a field if $n \geq 2$?

Show that $\mathbb{Z}/(p)^n$ does not contain a field if $n \geq 2$ and cannot be made into a vector space over $\mathbb{F}_p$ in a compatible way with its Abelian structure.
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Minimal polynomial: is $\sqrt[3]{2+\sqrt{5}}+\sqrt[3]{2-\sqrt{5}}=1$?

I was wondering about the minimal polynomial of real number $$u=\sqrt[3]{2+\sqrt{5}}+\sqrt[3]{2-\sqrt{5}}$$ over field $\mathbb{Q}$. As you can see here, I worked out that $u$ is a root of monic ...
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what is the definition of linearly independent subset of abelian group?

WHat is the definition of linearly independent subset of abelian group? I know the concept of this, but i don't know how to define this term precisely. Below is what i tried to formulate: Let ...
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44 views

Which non-Abelian finite groups contain the two specific centralizers? - part II

This is a question requiring the good knowledge of group theory: (Q1) Which finite groups $G$ contains some specific centralizers isomorphic to both of these two groups (but may contain other ...
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Which finite groups contain the two specific centralizers?

This is a question requiring the good knowledge of group theory: (Q1) Which finite groups $G$ contains some specific centralizers both of these two groups: i. the elementary group $Z_2^4$, ...
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Subgroup of group of order $44$

Pick the correct statement(s) below: $(a)$ There exists a group of order $44$ with a subgroup isomorphic to $ Z_2 + Z_2 $. $(b)$ There exists a group of order $44$ with a subgroup isomorphic to $ ...
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To Find a subgroup of order $6$ in $U(700).$

Find a subgroup of order $6$ in $U(700).$ Attempt: $U(700)=U(2^2.5^2.7)\thickapprox U(2^2) \oplus U(5^2)\oplus U(7)$ $\thickapprox \mathbb Z_2 \oplus \mathbb Z_{5^2-5} \oplus \mathbb Z_{7-7^0}$ ...
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normal group $(\text{ker }g)(\text{ker } f)$ as a kernel of some group using $f$ and $g$

For group homomorphism $f: A\to B$ and $g: A\to C$ we know $\text{ker }f\cap \text{ker } g$ is kernel of $(f,g): A\to B\times C$. $(\text{ker }g)(\text{ker } f)$ is trivially normal in $A$, can we ...
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Showing that a set is Discrete Valuation Ring

An order function on a field $K$ is a function $ϕ:K→Z∪{∞}$ satisfying: i) $ϕ(a)=∞$ if and only if $a=0$. ii) $ϕ(ab)=ϕ(a)+ϕ(b)$. iii) $ϕ(a+b)≥min(ϕ(a),ϕ(b))$. Show that R=$\{z∈K∣ϕ(z)≥0\}$ is a DVR ...
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48 views

Subfield of the Galois Group of $x^5 - 1$

Why is it that the subfield fixed by the subgroup of this Galois group is $\mathbb{Q}(\sqrt5)$. Can someone explain it without using the cyclotomic extension of $\mathbb{Q}$? Thank you edit: Using ...
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Separable and Splitting fields

Does a separable field imply splitting field? I am thinking yes. For $F < E$ If every $\alpha \in E$ is separable over $F$ then it must be that $E$ is a $S.F.$ over $F$? Also we have the ...