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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Verifying formulas for reflection, rotation, and translation in the Complex plane.

1) If $T_a : \Bbb C → \Bbb C$ is given by $T_a(z) = z + a$ then $T^{−1} _a (z) = z − a = T_{−a}(z)$ for some fixed $a ∈ \Bbb C$. 2) If $ρ_θ : \Bbb C → \Bbb C$ is given by $ρ_θ(z) = e^{iθ}z + a$ then ...
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HW - Number of subspaces T of a vector space K containing a fixed subspace M.

Given a vector space $K$ of dimension $k$ over a finite field $\mathbb{F}_q$, what is the number of subspaces $T$ of dimension $t<k$ that contain a given subspace $M$ of dimension $m<t$? ...
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For which $n \in \mathbb{N}$ does $x^8 + [2]_nx^7+[6]_nx^6-x^2-[2]_nx - [6]_n = [0]_n$ have at least $7$ distinct solutions?

I have to find one $n \in \mathbb{N}$ such that $$x^8 + [2]_nx^7+[6]_nx^6-x^2-[2]_nx - [6]_n = [0]_n$$ has at least $7$ distinct solutions in $\mathbb{Z}_n$ (or, equivalently, $f(x) = x^8 + ...
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If $G$ acts $k$-transitive and $k > 5$ and $G$ is neither alternating nor symmetric, then $(n-k)! \ge 2n$

The following is an exercise from D. Robinson: A Course in the Theory of Groups. Let $G$ be a $k$-transitive permutation group of degree $n$ which is neither alternating nor symmetric. Assume $k ...
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24 views

Centralizer of $(1,2,3)(4,5,6,7,8) \in A_{11}$

Let $$\tau = (1,2,3)(4,5,6,7,8) \in A_{11},$$ where $A_{11}$ is the group of even permutations. Let $H$ be the centralizer of $\tau$. I can easily prove that $H$ is a subgroup of $A_{11}$. How do I ...
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48 views

For which $p$ is $G_p = \{\sigma \in S_{12}: \text{ord}(\sigma) \text{ divides } p \}$ a subgroup of $S_{12}$?

Let $p$ be a prime number. I've to figure out for which $p$ $$G_p = \{\sigma \in S_{12}: \text{ord}(\sigma) \text{ divides } p \}$$ is a subgroup of $S_{12}$. I realize that we have to ...
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1answer
40 views

Group isomorphism between two groups .

Let $SO_2(\mathbb{R})$ be the group of rotations of the circle under the operation of composition. And consider $\mathbb{R}$ as an additive group. Prove that $SO_2(\mathbb{R}) \cong \begin{Bmatrix} ...
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42 views

Local properties of morphisms of schemes

In Hartshorne Proposition II.5.8, he shows, given a morphism $f \colon X \to Y$ where X and Y are schemes and $\mathcal{G}$ a quasi-coherent sheaf of $\mathcal{O}_{Y}$- modules, that ...
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41 views

Alternate proof to number of monomials in a given degree - “more” rigorous, formal [duplicate]

Let $s$ be the number of variables and $n$ be the degree of the monomials we want to count in $R[X_1,\dots,X_s]$. Then show, that the count is $$\delta(n,s):=\binom{s-1+n}{s-1}.$$ The question ...
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78 views

Understanding the definition of the sign of a permutation , $\operatorname{sgn}(\pi) = (-1)^k$ .

I am trying to understand the definition of the sign of a permutation $\pi$. My textbook only mentions that $\operatorname{sgn}(\pi) = (-1)^k$ , where $k$ is the number of transpositions . But I ...
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1answer
23 views

Studying the symmetry group of the dodecahedron by introducing axes.

For my bachelor project I'm studying the symmetry of the Platonic solids (as a start at least). I computed the symmetry group of the tetrahedron by labeling the vertices, and the cube by labeling the ...
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20 views

Degree of field extension using minimum polynomial

Let $K \subset L$ an algebraic extension. I want to prove for $a,b \in L$ that $$[K(b): K] \geq [K(a,b): K(a)],$$ Where for example $[K(b):K]$ is the degree of the field extension $K \subset K(b)$. ...
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21 views

Proving an example on Sum Of Ideals [on hold]

Show that $A$ is an ideal of $A+B$ Show that $A+A=A$ for some ideal $A$ in a ring $R$
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47 views

subgroup having index $2$ of $R^*$

The question is to find all the subgroups of $R^*$ (non-zero reals under multiplication) of index $2$. The index can be found out for finite groups. How to find subgroups having certain index for an ...
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38 views

Finding subgroups with a specific property.

I'm trying to find subgroups with the following properties: if $[G:H]=n$ there is a $g∈G$ so that $g^n≠e$. What do I do know is that $H$ cannot be normal (previous exercise). I just can't find any ...
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$\Bbb R,\Bbb C,\Bbb H$ are the only complete normed division rings

Does anyone have a proof or reference for this often-quoted fact? How complicated is the proof? If $K$ is a complete normed division ring, then $K$ is isomorphic to the real numbers, complex ...
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1answer
33 views

Two-dimensional algebras over complex numbers

I have read that there are two 2-dimensional algebras over the complex numbers, but I wanted to see what they are and how they are formed. I understand how to show that there are three 2-dimensional ...
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1answer
21 views

Proving that divisibility in an integral domain is a partial ordering

Given that R is an integral domain. I'm trying to prove that divisibility on this set constitutes a partial ordering. In particular, I have defined the relation $y \leq_{\,d} x$ on R by $y|x$. ...
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31 views

If $H \triangleleft K \leqslant G$, which requirements must be placed on $K$ in order to obtain $N \triangleleft G$?

Let $H$ be a normal subgroup of $K$, which is a subgroup of $G$. By just sketching some computations it seems that $H$ is not necessarily a normal subgroup of $G$ even if $K$ is a normal subgroup of ...
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35 views

how to normalise these values

First of all, i don't know if the correct word is normalise or not, but I'll try to explain my issue. I have a relationship between an object A and an object ...
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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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78 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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21 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
57 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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42 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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151 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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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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30 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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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
49 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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40 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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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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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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Show that Pn is an (n+1)-dimensional subspace [closed]

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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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 ...
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1answer
43 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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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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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
28 views

Ideals and submodules are the same [on hold]

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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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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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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1answer
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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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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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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 ...
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1answer
31 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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38 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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17 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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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
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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 ...
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1answer
29 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 ...