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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understanding the commutator of dihedral group [duplicate]

Let $G=D2n=⟨x,y|x^2=y^n=e, $ $yx=xy^{n-1}⟩$ i need to find G' [ the commutattor of G] now i understand the G' is the subgroup that is generated from $ U=xyx^{-1}y^{-1} , $ $\forall x,y \in G$ so ...
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isomorphic group [closed]

Pick the correct statement(s) below. (a) There exists a group of order $44$ with a subgroup isomorphic to $\mathbb{Z}/2 ⊕ \mathbb{Z}/2$. (b) There exists a group of order $44$ with a subgroup ...
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97 views

Which of the following abelian groups are cyclic groups?

Given the abelian groups of order $7425$: $$Z_{33} \times Z_{15} \times Z_{15} , \ Z_{25} \times Z_{297} , \ Z_{45} \times Z_{165} , Z_{55}\times Z_9 \times Z_{15}$$ Which of these groups, if ...
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30 views

Determine which roots of unity have degree at most 3

I need help to do exercises on "Abstact Algebra": 1.Determine all integer $n$, such that $\phi_n$ has degree at most $3$ over $\mathbb Q$, where $\phi_n=e^\frac{2\pi i}{n}$ .
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Doubly transitive [closed]

Show that $S_n$ acts doubly transitively on $X = \{1, 2, \dots, n\}$. Show that $D_{2n}$ acts transitively but not doubly transitively on the vergices of the regular $n$-gon for $n \ge 4$. Show that ...
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1answer
28 views

Infinite Varieties and Non constant Common Factors

I'm trying to work out some problems from Ideals, Varieties, and Algorithms, and I've stumbled on one that I'm unsure of how to start: Let $f,g \in \mathbb{C}[x,y]$ be nonzero. In this exercise, ...
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symmetry group of hypercube in $\mathbb{R}^4$ [closed]

Let $ = \{(x, y, z, w) \in \mathbb{R}^4 \text{ }|\text{ }|x|,\, |y|,\,|z|,\,|w| \le 1\}$ be the hypercube in $\mathbb{R}^4$ of side length $2$ centered at the origin. Identify the symmetry group of ...
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37 views

let $G$ to be finite abelian group of order O(G), let n to be prime number and (O(G),n)=1 prove that $g=x^n$ for any $ x \in G$

let $G$ to be finite abelian group of order O(G), let n to be prime number and (O(G),n)=1 prove that $\forall g \in G$ we can write $g=x^n$ for any $ x \in G$
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53 views

“Non-linear” algebra

Linear algebra studies vector spaces and linear mappings between those spaces. What tools do we use for NON-linear mappings between vector spaces?
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32 views

Questions about the dihedral group $D_8$ [duplicate]

Consider the dihedral group $D_8$ of order $16$. Consider $D_8$ with the presentation $D_8=\{r^i s^j : i=0,...,7; j=0,1; r^8=s^2=e; sr=r^7s=r^{-1}s\}$, where $\{e\}, \{rs, r^3s, r^5 s, r^7s\}$ and ...
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34 views

for $n \ge 3$, $S_n$ is isomorphic to its group of inner automorphisms

How would I go about showing that for $n \ge 3$, $S_n$ is isomorphic to its group of inner automorphisms?
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3answers
30 views

let $G$ be dihedral group so that $D_{2n}= \langle x,y \ | \ x^2=y^n=e, yx=xy^{n-1} \rangle$. Prove that $N\unlhd G$ and $G/N \cong W = \{1,-1\}$

let $G$ be dihedral group so that $G = D_{2n}=\langle x,y \ | \ x^2=y^n=e, yx=xy^{n-1} \rangle$. 1) let $N < G$ so that $N =\langle y \rangle = \{e,y,...y^{n-1}\}$. Prove that $G \unlhd N$. 2) ...
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29 views

Thompson Subgroup [closed]

A Thompson subgroup $J(G)$ is the subgroup of $G$ generated by all abelian subgroups $A$ of $G$ with maximal rank. Prove that $J(G)$ is a normal subgroup.
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26 views

Abelian Subgroups of $Q_8$ and $A_4$

What are the abelian/commutative subgroups of the quaternion group of order 8 and the alternating group? How does one go about in determining this? Are there theorems that relate normal subgroups to ...
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9 views

Minimal Number of Generators for Commutative Subgroups of G

A subset $S$ is said to generate a group $G$ if every element of $G$ can be written as a finite product of elements of $S$ and their inverses. Let $d(G)$ be the minimum number of generators of $G$. ...
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24 views

If $x$ is an integer and $m$ is an element from a ring prove that $x(-m) = -(xm)$

So my approach to this was to break it into 3 cases. Where x is =,>,< 0. The cases where x = 0 and x > 0 are easy. But I'm struggling with x < 0. Here is what I have so far: Let $x = -y$ ...
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1answer
50 views

Definition of division algebra

The definition on Wikipedia of a division algebra $D$ is given as: Given $a,b \in D$, $b \neq 0$ there exists a unique $c\in D$: $a = bc$ and a unique $d \in D$: $a = db$. My question(s) are: What ...
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43 views

Application of Chinese remainder theorem

from Chinese remainder theorem, we know that of $m,n \in \mathbf{Z}$, $(m,n) = 1$, then $Z_m \otimes Z_n \cong Z_{mm}$ as ring isomorphism, but how it related to the application that $\phi(nm) = ...
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21 views

well-defined action, injective homomorphism, action of $G$ on set of pairs of opposite faces of cube

Let $X = \{1, \dots, n\}$. Let $G$ act on $X$, denoted by $g \cdot x$ for $g \in G$ and $x \in X$. Let $K$ be the kernel of this action. Show that the map $(gK) * x = g \cdot x$ gives a well-defined ...
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19 views

Minimum Number of Generators

A subset $S$ is said to generate a group $G$ if every element of $G$ can be written as a finite product of elements of $S$ and their inverses. Let $d(G)$ be the minimum number of generators of $G$. ...
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if $m, n \in \mathbb{N}$, $m < n$, then $S_m$ isomorphic to subgroup of $S_n$

How do show that if $m, n \in \mathbb{N}$ and $m < n$, then $S_m$ is isomorphic to a subgroup of $S_n$, without using any "overpowered" results?
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group iff $G$ is a set, $*$ associative binary operation satisfying left identity, left inverse

How would I go about showing that the pair $(G, *)$ is a group if and only if $G$ is a set and $*$ is an associative binary operation on $G$ such that: (Left Identity) There exists an element $e \in ...
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29 views

Prove that an exact sequence splits

Let $0 \to r\mathbb{Z}_n \to \mathbb{Z}_n \to s\mathbb{Z}_n \to 0$ where n =rs an exact sequence of $\mathbb Z$ modules the how can I prove the sequence splits if and only if $(r,s)=1$ The only thing ...
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45 views

What are the elements of $Z[x]/\langle x^2 - 1\rangle$

I have a hard time visualizing what the actual elements of this ring are. Because it is mod the elements generated by $x^2 - 1$ are polynomials with non-even powers in the ring or not?
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38 views

Find the cardinality of the group $GL(2, \mathbb{Z/p^{n}Z}) $

Find the cardinality of the group $GL(2, \mathbb{Z/p^{n}Z}) $ for each prime $p$ and positive integer $n$ . What I know : Clearly if $n=1$ then cardinality of $GL(2, \mathbb{Z/pZ}) $is number of ...
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107 views

Why can a matrix whose kth power is I be diagonalized?

Say A is an n by n matrix over the complex numbers so that A raised to the kth power is the identity I. How do we show A can be diagonalized? Also, if alpha is an element of a field of characteristic ...
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Semisimplicity of the ring $\mathbb Z_n$

I am being asked to figure out when $\mathbb Z_n$ is a semisimple ring. It is clear to me that if $n$ is prime then $\mathbb Z_n$ is simple, which implies it is semisimple. If $n=p_1...p_n$ is a ...
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Simple problem about morphism in abelian categories

$f$ : $X\to$ $Y$ and $g$ : $Y\to$$Z$ a sequence in abelian categories. Show that if $gf$=$0$ if and only if exist a monomorphism $h$:$Im(f)$ $\to$ $Ker(g)$ such $kh$=$j$, where $j$:$Im(f)$$\to$ $Y$ ...
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Showing product of disjoint cycle

I am trying to show the product of two disjoint cycles such that they have nothing in common for $A_n$ for $n\ge 3$. So I have the two cycles $(ab)(cd)$. I have read here: ...
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Eigenspaces and maximality

QUESTION: Given that $V$ is a vector space. And $V=\bigoplus_i E_i(\lambda_i)$. Where $E_i(\lambda_i)$ are the eigenspaces of $V$. Is it true that given any $T$ invariant subspace $U$ of $V$ then $U$ ...
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Eigenvectors and Generalized Eigenvectors

I've wondered whether someone could calrify me what are Generalized Eigenvectors, and why can I use them to find triangular form of a matrix. Say I have a $3\times3$ matrix, and I want to bring it to ...
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Why $\mathbb{Z}/n\mathbb{Z}$ isn't a subgroup of $\mathbb{Z}$

Could anyone explain to me why this isn't true? It's listed as an example in our textbook but no reason is given. I've checked the properties of a subgroup, and it seems to follow them. What am I ...
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If $A$ is isomorphic to $B$ and $B$ is a field, then $A$ is a field?

This is a follow-up question to Let $f$ be a surjective homomorphism. Prove that $\ker(f)$ is a maximal ideal. In that question, we know that since $S$ is a field and $R/\ker(f)$ is isomorphic to $S$ ...
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1answer
54 views

What ring is isomorphic to factor-ring?

$R=\mathbb{Z}[x]$ - polynomials with integer coefficients $I = (x^2+x-1)\mathbb{Z}[x]$ In this case we have that classes of factor-ring $R/I$ represented in the next form: $K_{a,b}=ax+b$. What ring ...
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The probability of modulo a prime

Suppose i have a uniform random number generator which generates integers uniformly over some range [x,y] The output obtained z, can be binned into p buckets via: z mod p if p were prime, are the ...
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Proof that the ideal of set of polynomials is generated by its gcd

Theorem: Given $\{f_i\}_{1 \leq i \leq n}$, $f_i \in \mathbb{K}[x]$. Then the monic generator $f$ of the ideal $\langle \{f_i\} \rangle$ is $f = \gcd \{ f_i \}$. In other words: $\langle \{f_i\} ...
3
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1answer
51 views

Polynomial Pell equation

Can someone point me in the right direction? Let $k$ be a field of characteristic $0$ and let $D \in k[x]$ be non-constant. Prove that the ‘polynomial Pell’ equation $$f^2 − Dg^2 = 1,\,\,\,\,f, ...
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46 views

Permutation order

How do you put non disjoint permutation cycles into disjoint cycle form? For Example the permutation in non disjoint cycle form (1352)(34)? How do you form disjoint cycle for from this?
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Different forms for the exterior power of a module

First I have defined the exterior algebra of a module $M$ as the quotient $T(M)/A(M)$ where $T(M)$ is the tensor algebra of $M$ and $A(M)$ is the ideal generated by all elements of the form $m\otimes ...
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Proving equivalent versions of faithfully flatness.

I was reading a proof of the the following theorem from Matsumura (p.47) There was something confusing about $(3) \implies (2)$ and $(2) \implies (1)$. Question 1 Here, it says $M \not= ...
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Showing that $A^*$ is and ideal and that “$*$” is multiplicative.

Let $E/F$ be an extension of algebraic number fields and $\mathcal{O}_E$ and $\mathcal{O}_F$ be the ring of integers. Define $$A^*=\mathcal{O}_EA,$$ where A is an ideal of $\mathcal{O}_F$. Prove that ...
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Infinite Units for $\mathbb{Z}[\sqrt{7}]$

Suppose that $\alpha \in \mathbb{Z}[\sqrt{7}]$ where $\alpha$ is of the form $a + b\sqrt{7}$ where $a, b \in \mathbb{Z}$. Because $\alpha$ is a unit if and only if $N(\alpha)=\pm 1$ we must show: ...
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1answer
13 views

Question about field extensions regarding minimal polynomial of multiple of algebraic element

Let $K/F$ be a field extension, let $α ∈ K$ be an algebraic element with minimal polynomial $f(X) ∈ F[X]$, and let $r ∈ F^\times$. What is the minimal polynomial for $rα$ in terms of $r$ and $f$? I ...
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+100

Factorizations in the symmetric group

Notation notes: The cycle $(i,i+1,\dots,j)\in S_n$ is the permutation $(1)(2)...(i,i+1,\dots,j)\dots(n)$ in cycle notation. Motivation Given a factorization of a permutation into certain cycles ...
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2answers
46 views

A group in between the commutator subgroup and the original group must be normal

Let $C$ be the commutator subgroup of a group $G$. then by some easy arguments, we know that $1$. $C$ is normal in $G$ $2$. $G/C$ is abelian $3$. If $N$ is normal in $G$ and $G/N$ is abelian, then ...
3
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3answers
120 views

A sequence of roots of polynomials depending on an integer parameter

For $n\in \mathbb N-\{0\}$, let $$Q_n=(2n-1)X^n+(2n-3)X^{n-1}+(2n-5)X^{n-2}+\cdots+3X^2+X$$ I want to show that there is a unique $x_n\geq 0$ such that $Q_n(x_n)=1$ and then show that the sequence ...
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1answer
38 views

Cryptography question

I think this is a pohlig hellman symmetric key system working in $\mathbb{Z}/p\mathbb{Z}$ Assuming Alice and Bob both have p (a prime) and k (a key) If Alice sends $m^k$ to Bob, can Bob raise $m^k$ ...
3
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1answer
36 views

Finite abelian groups (application of structure theorem)

Problem Find all finite abelian groups that simultaneously have exactly $7$ elements of order $2$, exactly8 elements of order $3$, exactly $8$ elements of order $4$, at least an element of order ...
2
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1answer
28 views

Direct limits in the category of modules

STATEMENT: Proof. Let $(I,≼)$ be a directed set, and let $\left\{Mi\right\}i∈I$ be a directed system of R-modules, with $\left\{f_{ji}\right\}_{i∈I},i≼j$ a corresponding directed family of ...
5
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1answer
47 views

Group of order $p^{n}$ has normal subgroups of order $p^{k}$

Q: Prove that a subgroup of order $p^n$ has a normal subgroup of order $p^{k}$ for all $0\leq k \leq n$. Attempt at a proof: We proceed by Induction. This is obviously true for $n=1, 2$. Suppose it ...