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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silly confusion about subgroup proof: $< g,h> = \{g^r h^s : r, s \in Z\}$

Could someone help me understand this? It says certainly $<g,h> \supseteq \{g^r h^s : r, s \in Z\}$ which i understand. But now arent we suppose to show that $<g,h> \subseteq \{g^r h^s : ...
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If a polynomial $g$ divides $f$ and $f'$, then $g^2$ divides $f$?

Here's a homework problem from Artin's Algebra that I'm having a lot of trouble with Let $f(x) \in F[x]$ (where $F$ is a field of characteristic $0$). If $g$ is an irreducible polynomial that is ...
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proof verification: If $f:G \rightarrow H$ is group homomorphism, and $H$ is abelian, then $G$ is abelian

If $f:G \rightarrow H$ is group homomorphism, and $H$ is abelian, then $G$ is abelian. Is that statement correct? Here's my attempt of proof: Let $a,b \in G$, then: ...
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1answer
14 views

Question on Cardinality ..Help

a) Let $n$ be a positive integer. Define a relation on $\mathbb{Z} $, which yields a partition of $\mathbb{Z}$ with $n$ elements; and give the partition. b) Deduce that $n\omega = \omega$ where ...
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1answer
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Roots of monic polynomials

I'm trying to show that if $\alpha$ is a rational root of $a_nx^{n}+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$, then $a_n \alpha$ is a root of monic ...
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2answers
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Supose that G is a finite abelian group that does not contain a subgroup isomorphic to $Z_p \oplus Z_p$ for any prime $p$. Prove that G is cyclic.

Supose that G is a finite abelian group that does not contain a subgroup isomorphic to $Z_p \oplus Z_p$ for any prime $p$. Prove that G is cyclic. Attempt: If $G$ is a finite abelian group, then let ...
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Showing intersection of two finite-indexed groups is finite

Let $H, K$ be subgroups of $G$ with finite indexes, and $K\lhd G$, $H\lhd G$. Show $H \cap K$ has finite index. We were taught only first and second homomorphisms theorems, and not all the indexes ...
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34 views

Showing to be nonabelian group

Show that $S_n$ is nonabelian group for $n≥3$. How can we show this? What are the conditions of being a nonabelian group? Thank you.
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28 views

Equivalence relation and equivalence class question

Show that the relation $\sim$ defined on the set $X = \mathbb{N} \times \mathbb{N} = \{(a, b) : a \in \mathbb{N}; b \in \mathbb{N}\}$ as $(a,b) \sim (c,d)$ if and only if $a + d = c + b$ is an ...
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1answer
30 views

Question about Quaternion group $Q_8$ and Dihedral group $D_8$

pretty much got stuck with the following question (it has several parts): a). Show that $D_8$ isn't isomorphic to $Q_8$ b). Let $K$ be a subgroup of $GL_2(\mathbb C)$ so that $$K=\left\langle ...
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1answer
19 views

Subgroups of $ D_8 $

$D_8 =\{ e, r, r^2, r^3, s, rs, r^2s, r^3s\} $ where r is a rotation by $ \frac{\pi } {2} $ anticlockwise and s is a reflection. Then according to my book, there are 10 subgroups, four of which are ...
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1answer
25 views

Primitive elements for $K=\Bbb{Q}(\sqrt{2},\sqrt{3})$

The key lemma for proving the primitive Element Theorem (for finite extension of a field $F$ with characteristic $0$) in Artin's Algebra (2nd edition) is the following: Suppose $char F=0$ and ...
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2answers
24 views

$(34)(123)(456)$ is a cycle. True or False?

I know this is basics, and I understand that $(34)(123)(456)$ is a product of cycles which, I found: $(124563)$. But somehow, I was lost. How do I know if it is indeed a cycle? OR if it isn't? Any ...
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0answers
9 views

puzzle on [13,10,3] perfect Hamming code over $\mathbb F_{3}$

The soccer betting form contains a list of 13 games. There are three possible outcomes for each game: “the first team won”, “the second team won” and “draw”. Each betting form allows to chose one ...
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8 views

Constructing $\mathbb{Z}$-bilinear map from $\mathbb{R} \times (\mathbb{R} / \mathbb{Z})$

I'm trying to show $\mathbb{R} \otimes_\mathbb{Z} (\mathbb{R} / \mathbb{Z})$ is nontrivial, where my tensor product is defined using the universal property. This problem can be easily reduced ...
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1answer
30 views

Verifying whether a quotient ring is indeed a ring.

Take $$\frac{\Bbb{R[x]}}{\langle x^2+1\rangle}$$ This is a ring. In this quotient ring, the product of equivalence classes $[a+bx]$ and $[c+dx]$ is another equivalence class, as a ring is closed under ...
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let $G$ be an infinite group of the form $G_1 \oplus G_2 \oplus \dots \oplus G_n$ , $n>1$. Prove that $G$ is not cyclic

let $G$ be an infinite group of the form $G_1 \oplus G_2 \oplus \dots \oplus G_n$ where each $G_i$ is a non trivial group and $n>1$. Prove that $G$ is not cyclic. Attempt : Let $G = G_1 \oplus G_2 ...
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2answers
33 views

Proof of a basic isomorphism.

How do you prove that $D_2 \cong V \cong \mathbb{Z}_2 \oplus \mathbb{Z}_2$? Where $V$ is the Klein-4 Group and $D_2$ is the dihedral group with cardinality 4. We have that $D_2 := \{1,r,s,sr \}$ ...
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17 views

tables of cyclic subgroups and conjugates

$G = S_5$, I need to construct tables for $H$ and $aHa^{-1}$ ($H =$ cyclic subgroup $(142)(35),$ and $a = (2354) \in G$) and see what can be inferred. In my attempt $H$ = $\{(142)(35), (124)(35), ...
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1answer
23 views

Group Theory proving [on hold]

can someone help me with this question? 1) Given a natural number n≥1, let $G_n$ be the set of complex n-th roots of $1$, i.e. $G_{n} = \{z \in \mathbb{C} :z^n = 1\}$ Prove that $G_n$ is a group ...
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If $|\Phi(A)|=|\Pi|$, then $O_{L}=o_{k}[A]$.

Here the problem: K/k is a finite algebraic extension, $\Pi$ is a prime element in K, $A\in O_{K}$, p is the maximal ideal of k and P the one for K. We have that $\bar{A}:=A mod P$ generates the ...
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If $a\equiv b [p^k]$ then $a^p \equiv b^p [p^{k+1}]$

Can anyone explain the steps to this proof? I'm really lost/ If $k\geq 1$ and $a\equiv b[p^k]$ then $a^p \equiv b^p [p^{k+1}]$ Proof: Since $a= b + qp^k$ for some $q\in \mathbb{Z}$ we have $a^p = ...
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0answers
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Is $L:=\mathbb{Q}_{2}[\sqrt{1+\sqrt{-2}}]$ Galois extension of $\mathbb{Q}_{2}$? completely ramified?

I think it is because normal: The minimal polynomial is $f(x)=(x^{2}-1)^{2}+2$ which has roots $1,\sqrt{-2}, \sqrt{1+\sqrt{-2}},\sqrt{-2}\sqrt{1+\sqrt{-2}}$ and those are all contained in L. ...
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Showing that no non-identity element of $G/F_g$ has finite order where $G$ abelian, $F_g$ the set of elements of G that have finite order

Let $G$ be an abelian group and $F_g$ the set of elements of $G$ that have finite order. Show $F\trianglelefteq G$ and no non identity element of $G/F_g$ has finite order. $G$ is abelian $\implies ...
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2answers
52 views

Why can't we have an $y$ such that $xy\equiv 1\; (mod\; n)$ when $n$ is not prime?

I'm reading Avner's Fearless Symmetry: Here he says that we can only have the cancelation law if the modulus is prime: I got curious with the statement and then I kept reading the chapter: ...
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68 views

Algebra, Geometry and Algebraic Geometry

I want to know, what is the difference between Algebra, Geometry and Algebraic Geometry ? Your reply is highly appreciated.
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Image of the Brauer group under a field extension

For $k$ a field, let $Br(k)$ - the Brauer group of $k$ - denote the group of finite-dimensional central simple algebras over $k$, modulo Morita equivalence $(A\equiv B\iff \exists m, n(A\otimes_k ...
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Normalizer of a subgroup of $GL_2(\mathbb{R})$

I have following subgroup of $GL_2(\mathbb{R}):$ $$A=\Bigg\{\left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right),\left( \begin{array}{cc} 0 & -1 \\ 1 & 0 \end{array} ...
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1answer
27 views

Conjugacy classes and centralizers of a SmallGroup

What is the complete lists of conjugacy classes and centralizers of SmallGroup(64,138)? Would someone be willing to provide the complete lists of conjugacy classes and centralizers of ...
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45 views

Frobenius number and Arf ring of a semigroup [on hold]

Let $$G=\{5m+7n \mid m, n\in \Bbb N\}.$$ 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 ...
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1answer
29 views

$\exists a\in G-H$ 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
22 views

Notation question: Group generated by two elements?

Let there be $H$ subgroup of symmetric group $S_4$, so that $H= \langle (12)(34),(234) \rangle$. What does the notation $\langle (12)(34),(234) \rangle$ mean? I know that if there's one elements, then ...
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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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0answers
62 views

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
72 views

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
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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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36 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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$Z$ 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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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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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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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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12 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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50 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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25 views

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
16 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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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
25 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 $4n+2$.
3
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3answers
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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 ...