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

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multiplication of quaternions is like complex numbers multiplication?

Suppose $p = z + j w $ where $z = x_0 + i x_1$ and $w = x_2+ix^3$. Let $q = \alpha + j \beta $ where $\alpha = y_1 + i y_2$ and $\beta = y_2 + i y_3$. How can we multiply $p$ and $q$. Is is just like ...
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0answers
20 views

Injectivity in the zero homology

I'm struggling with following step in an excercises about Mayer-Vietoris sequences: In one step the solution says this map is injective since $A \cap B$ is path-connected: $$ H_0(A \cap B) ...
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0answers
19 views

Reciprocal of a quaternion in matrix form

I am given the definition that $$ q\longleftrightarrow \left[ \begin{array}{cc} z&w\\ -\overline{w}&\overline{z} \end{array} \right], q = z + j \overline{w}, z = x_0 + i ...
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1answer
37 views

Proper subgroup of $S_{15}$ that strictly contains $\sigma $

How does one prove that: There exists no cyclic proper subgroup of $S_{15}$ that strictly contains the following permutation (of order $10$)? $$\sigma = (1, 2,3, 4,5, 6, ...
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2answers
30 views

Examples of fields with characteristic $2$. [on hold]

What are good examples of fields of characteristic $2$, starting from the simplest one to more interesting examples?
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2answers
58 views

Example of a Tensor Product of Modules with Non-Decomposable Elements

Given a ring $R$ and $R$-modules $A_R$ and $_{R}B$, we define the tensor product $A \otimes_R B$ as the free abelian group on $A \times B$ modded out by the subgroup generated by the elements of ...
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4answers
134 views

Definition of Inverse in Linear and Abstract Algebra

In a linear algebra text, the following is the definition of the inverse of a matrix An $n\times n$ matrix $A$ is invertible when there exists an $n \times n$ matrix $B$ such that $$AB = BA = ...
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0answers
25 views

Intersection of subgroup with Sylow subgroup [on hold]

Let $p$ be a prime number. If $P$ is sylow $p$ subgroup of $G$ of some finite group $G$ then for every subgroup $H$ of $G$, $H \cap P$ is $p$ sylow subgroup of $H$? True of false?
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2answers
20 views

what is the difference between finitely generated module and finitely generated free module?

I am still confused about the difference between free module and finitely generated module. For example, $Z/2Z$ is finitely generated module, but why it is not finitely generated free module? What is ...
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1answer
32 views

Correspondence between prime and maximal ideals [on hold]

My professor put the following statement in the lecture notes without proof: Let $R$ be a commutative ring and $I$ an ideal. Then the natural correspondence between ideals containing $I$ and ideals ...
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2answers
34 views

Linear transformation from a vector space to a field

Can anyone help me with the following question: Let $V$ be a vector space over field $F$, possibly not finite dimensional. Let $T \colon V \to F$ be a linear map. Prove that there is no subspace ...
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2answers
68 views

Is it true that $\mathbb{Q}(\sqrt{2}) \cap \mathbb{Q}(i) = \mathbb{Q}$?

Is it true that $\mathbb{Q}(\sqrt{2}) \cap \mathbb{Q}(i) = \mathbb{Q}$? I know that \begin{align*} \mathbb{Q}(\sqrt{2}) &= \{a+b\sqrt{2} \mid a,b \in \mathbb{Q}\}, \\ \mathbb{Q}(i) &= ...
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3answers
34 views

The union of three subspaces equals to a vector space

I am aware that the union of subspaces does not necessarily yield a subspace. However, I am confused about the following question: (i) Let $U, U'$ be subspaces of a vector space $V$ (both not ...
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1answer
27 views

Show that Pn is an (n+1)-dimensional subspace [on hold]

Show that $P_n = \{$Polynomials with real coefficients of degree $≤ n\}$ is an $(n+1)$-dimensional subspace of the infinite-dimensional vector space of all real polynomials.
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1answer
35 views

If a group has no subgroups other than the identity and itself, then it is finite and is of prime order [duplicate]

I want to prove that if a group has no subgroups other than the identity and itself, then the order of the group is a prime number. A hint would be appreciated. Is there any theorem on the relation ...
1
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1answer
32 views

Intersection of two ideals

Let $A$ be a commutative ring and let $\mathfrak{a}$, $\mathfrak{b}$ be ideals in $A$. I am asked the following question: Show that $\mathfrak{a} \cap \mathfrak{b}$ is the largest ideal of $A$ ...
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0answers
13 views

What does it mean that “$f = g$ in $k(h(t))$?”

Let $k$ be a field and consider the rational function field $k(t)$. I was just reading that if $f(t),g(t),h(t)\in k(t)$ are such that $f(h(t)) = g(h(t))$, then "$f = g$ in $k(h(t))$." What does that ...
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1answer
20 views

Showing cyclic group

Is $\Bbb R_{+}$ under multiplication cyclic? Would it suffice to show that this group is isomorphic to a cyclic group? For example, $\Bbb Z_+$ is cyclic under addition, $\Bbb R_{+}$ is isomorphic to ...
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1answer
23 views

Ideals and submodules are the same

My teacher has told me that for an R-module, that I is an ideal of R if and only if I is an R-submodule of R. I know this is true but I was wondering why? IS there an official proof?
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4answers
39 views

Showing the group in $\Bbb R$

I have the following problem I am confused about: Let $x,y\in \Bbb R, x\ast y=xy+x+y$ Is $\Bbb R$ a group? I wrote $(x\ast y)\ast z= x\ast (y\ast z)$, then calculated it, associativity did not hold. ...
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1answer
17 views

A question concerning isometries and determinants

Let $e_1 = (1, 0)$ and $e_2 = (0, 1)$. Assume that $f : \Bbb R^2 → \Bbb R^2$ is an isometry of the plane fixing $(0, 0)$. Let $f(e_1) = (a, b)$ and $f(e_2) = (c, d)$, and let $A = \begin{vmatrix} ...
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0answers
9 views

Algorithm for computing the inverse limit of a finite inverse system

Let $k$ be a field (finite if you'd like), let $(I,\le)$ be a finite directed poset with $|I|=n$, and let $(A_i,f_{ij})_{i\le j\in I}$ be an inverse system of finitely generated, graded, commutative ...
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0answers
28 views

An algebra exercise

Let $\sigma$ be a permutation. Prove that there exists some permutation $\rho$ such that $\sigma^{-1} = \rho \sigma \rho^{-1}$ I tried playing with it a little but to no avail.
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0answers
13 views

Induced homomorphism between quotient modules

Let $R$ be a commutative ring with identity. Let $M,N$ be $R$-modules, and let $I$ be an ideal in $R$. Let $\varphi : M \to N$ be a homomorphism such that the induced homomorphism $$\bar{\varphi} : ...
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0answers
12 views

Algorithm for computing an inverse image

Let $k$ be a field (finite if you'd like), and let $f:A\to B$ be a map of graded, commutative $k$-algebras. Suppose further that $A$ is finitely generated and choose a presentation ...
1
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1answer
24 views

Proving that a group representation is *not* a direct sum of irreducible represenations.

Problem Statement: Let $x$ be a generator of a cyclic group $G$ of order $p$. Sending $x\mapsto \begin{bmatrix} 1 & 1\\ 0 & 1 \end{bmatrix}$ defines a matrix representation $G\rightarrow ...
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1answer
32 views

Subgroup H group G.

Let $G$ be a group, $H$ a subgroup of $G$, and $N:=\cap_{x\in G} \ \ x^{-1}Hx$. Prove, that $N$ is normal subgroup in $G$. I did this: Let $g\in G$. Whether $g(xhx^{-1})g^{-1}\in N$? Take $f=gx$. ...
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0answers
16 views

Isomorphism of a quotient group [duplicate]

I have that $G=S_4$ and $N = \{1, (12)(34), (13)(24), (14)(23)\}$, and thus far I have shown that N is a normal subgroup of G. I'm trying to figure out what group $G/N$ will be isomorphic to, but I ...
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0answers
18 views

Order of a certain finitely generated group

Suppose I am looking at the group $W=\langle s_\alpha, t_\beta \rangle$ where $s_\alpha$ and $t_\beta$ are reflections in $\mathbb{R}^2$ coming from two vectors $\alpha$ and $\beta$ making an angle of ...
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2answers
18 views

Is the set of all $f$ such that $\lim_{x\to1^-}f(x) = 0$ an ideal in the ring of functions from $[0,1]\rightarrow \mathbb{R}$?

Is the set of all $f$ such that $\lim_{x\to1^-}f(x) = 0$ an ideal in the ring of functions from $[0,1]\rightarrow \mathbb{R}$? I'm sure about the closure under addition but not quite clear about if ...
2
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1answer
27 views

Integer (or whole) numbers in arbitrary fields.

Given an arbitrary field $K$, may I define an integer in $K$? I have found how to define an algebraic number in $K$ and how to define an integer algebraic number in $K$. For instance, let ...
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0answers
23 views

Suppose that $G$ is the disjoint union $G=\cup_{i=1}^n Sg_iT$.Prove that $[G:T]=\sum_{i=1}^n[S:S\cap g_iTg_i^{-1}]$ [on hold]

Let $S,T\leq G$,where G is a finite group, and suppose that $G$ is the disjoint union $$G=\cup_{i=1}^n Sg_iT$$ Prove that $[G:T]=\sum_{i=1}^n[S:S\cap g_iTg_i^{-1}]$ I don't have any idea how to start ...
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6answers
69 views

Prove or give a counterexample: If a group G has a subgroup of order n, then G has an element of order n

The question is given in the title. I am unable to come up with a counterexample and I'm thinking this could apply to cyclic groups but not necessarily to general groups. Does anyone have any ideas or ...
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1answer
35 views

For a, b ∈ R we define a ∗ b := ab − a − b + 2 ∈ R. Furthermore let G := R\{1}. [on hold]

For a, b ∈ R we define a ∗ b := ab − a − b + 2 ∈ R. Furthermore let G := R \ {1}. (a) Show that a ∗ b ∈ G for all a, b ∈ G. (b) Show that G together with the binary operation G × G → G, (a, b) |→ a ...
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1answer
24 views

Number of permutations with given cyclic structure

If $\sigma$ is a permutation made up by the disjoint cycles $\tau_1, \dots, \tau_r$ (including those of length $1$), we call structure of $\sigma$ $$(l_1, \dots, l_r),$$ where $l_1, \dots, l_r$ are ...
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0answers
30 views

Generalisation of euclidean domains

Recently I wondered how dependent the definition of euclidean domains is of the co-domain of the norm-function. To be precise: Let's define a semi-euclidean domain as a domain $R$ together with a ...
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1answer
19 views

Verify $(H_1 \cap H_2)g=H_1g \cap H_2g$ where $H_1,H_2\leq G$ and $g\in G$.

Verify $(H_1 \cap H_2)g=H_1g \cap H_2g$ where $H_1,H_2\leq G$ and $g\in G$. Let $S_1,S_2\leq G,g\in G$ What i had done, $x\in (S_1 \cap S_2)g$. Then $x=sg,s\in S_1\cap S_2$. So clearly, $x\in ...
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1answer
38 views

How to use the Universal Coefficient Theorem to determine $H^i(M; \mathbb{Z}_p)$ from $H^i(M; \mathbb{Z})$? [on hold]

Let $M$ be a path-connected finite $CW$-complex. Suppose $$ H^2(M;\mathbb{Z})=\mathbb{Z}_{2k}, \text{ } k\geq 3; $$ $$ H^3(M;\mathbb{Z})=\mathbb{Z}\times\mathbb{Z}_{2}; $$ $$ ...
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2answers
41 views

An irreducible polynomial in a subfield of $\mathbb{C}$ has no multiple roots in $\mathbb{C}$

Let $K\subset \mathbb{C}$ be a subfield and $f\in K[t]$ an irreducible polynomial. Show that $f$ has no multiple roots in $\mathbb{C}$. If I understand this question correctly, I must show that ...
2
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1answer
53 views

Number of solutions in a field of order $32$ [duplicate]

Let $F$ be a field of order $32$. Then find the number of non-zero solutions $(a,b)\in F\times F$ of the equation $x^2+xy+y^2=0$. As , $|F|=32$ , so $(F\setminus\{0\},.)$ forms a group of order $31$, ...
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0answers
37 views

Monoids and groups

everybody. I got this exercise from Jacobson. Let $M$ be a monoid generated by a set $S$ and suppose every element of $S$ is invertible. Show that $M$ is a group. Proof: every element of $M$ has ...
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1answer
46 views

Group of order 6 contains an element of of order 3

I need to show that if $G$ is non-abelian group of order $6$ then it contains an element of order $3$. I don't know how to proceed. Any kind of help/hint is appreciated.
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1answer
25 views

$f$ is divisible by a square of non-constant polynomial iff $f,f'$ are not relatively prime

Let $R$ be a commutative ring and $f=a_0+ \cdots +a_nt^n \in R[t]$. Define $f':=a_1+2a_2t+ \cdots + na_{n-1}t^{n-1}$. Show that $f$ is divisible by a square of non-constant polynomial if and only ...
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2answers
18 views

Union Over a Totally Ordered Set of Ideals is an Ideal

I am trying to understand the proof of a theorem which uses Zorn's lemma. I understand quite well all parts of the proof except for one point: Let $R$ be a ring and define $K\doteq \{I\subseteq R ...
1
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1answer
21 views

Characterization of simple representations

I am trying to solve exercise 10 from chapter 2 of Peter Webb's book on representation theory: Prove the following theorem of Burnside: let $G$ be a finite group and let $k$ be a algebraically ...
1
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1answer
33 views

Turning Preordered Sets into Preordered Monoids (Constructing Preordered Monoids from Preordered Sets)

Question: Referring to the Wikipedia article on Adjoint Functors in Section 2 (Motivation), they talk about "turning rngs into rings" (can be rephrased as "constructing rings from rngs"). I do not ...
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1answer
39 views

Determine all the subgroups of the dihedral group $D_{15}$

Is there an algorithm for finding all of the subgroups of $D_{15}$? Also, is there a formula for finding the size of that subgroup? Not sure where to start with finding all the subgroups of $D_{15}$ ...
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1answer
23 views

Finding an orthonormal basis for $\mathbb{C}^2$ with respect to the Hermitian form $\bar{x}^TAy$

Problem Statement: Let $A=\begin{bmatrix} 2 && 1 \\ 1 && 2 \end{bmatrix}$. Find an orthonormal basis for $\mathbb{C}^2$ with respect to the Hermitian form $\bar{x}^TAy$. I am ...
1
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1answer
35 views

Prove that the set $U = \{(123), (124), … , (12n)\}$ can be used to generate $A_n$.

A hint is provided with the proof prompt: $(abc) = (1ca)(1ab)$, $(1ab) = (1b2)(12a)(12b)$, and $(1b2) = (12b)^2$. My idea: $(1ab) = (12b)(12b)(12a)(12b)$. To solve for the other half of $(abc)$, I'm ...
-1
votes
1answer
38 views

showing non-isomorphism of groups [duplicate]

How do I prove that there is no isomorphism between $\Bbb Z$ under addition and $\Bbb Q$ under addition? They both are infinite order. I thought they might be isomorphic. Help would be ...