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.

learn more… | top users | synonyms (1)

0
votes
0answers
43 views

Show that $\phi(p^e)=p^e-p^{e-1}$

In an exercise I was asked to show that if $R$ is a ring with relatively prime ideals $I_1,I_2$ then $R/I \cong R/I_1 \oplus R/I_2$ where $I=I_1 \cap I_2$ and $\oplus$ is the direct sum. A follow on ...
1
vote
1answer
35 views

Name of and references for the equivalence relation $x \sim y :\Longleftrightarrow x^2 = y^2$

Playing around with the concepts of negativity and positivity, I came across the following equivalence relation defined for all elements $x,y$ of a field $\mathbb{F}$: $$ x \sim y :\Longleftrightarrow ...
1
vote
1answer
23 views

If I is an irreducible ideal, and P is a prime ideal, is (I+P)/P irreducible?

Let $A$ be a commutative ring with unit, and $P$ a prime ideal. My question is: If $I$ is an irreducible ideal in $A$, is $(I+P)/P$ irreducible in $A/P$? If not, can you show a counterexample? ...
0
votes
1answer
36 views

Prove that a group of infinite order must have a proper subgroup [on hold]

Assume that the group of infinite order is also cylic. How would one prove that? I am quite stuck.
0
votes
0answers
15 views

permutations and representations , sign function.

Show that the sign representation of $S_n$ is indeed a representation. attempt: Recall the sign function of a permutation is given by $\mathrm{sgn}(\pi) = (-1)^k$. Then recall a representation is a ...
4
votes
2answers
40 views

Roots of $x^p + x + [\alpha]_p \in \mathbb{Z}_p[x]$

Let $$g(x) = x^p + x + [\alpha]_p \in \mathbb{Z}_p[x],$$ where $p$ is prime. For which $\alpha, p \in \mathbb{Z}$ does $g(x)$ have at least one root? And for which $\alpha, p \in \mathbb{Z}$ ...
-2
votes
1answer
34 views

How can i prove that a group of order 60 Is not simple? [on hold]

How can i prove that a group of order 60 Is not simple? please help me guys . I m fed up to think about this
11
votes
9answers
781 views

Can an infinite sum of irrational numbers be rational?

Let $S = \sum_ {k=1}^\infty a_k $ where each $a_k$ is positive and irrational. Is it possible for $S$ to be rational, considering the additional restriction that non of the $a_k$'s is a linear ...
0
votes
1answer
18 views

$A_n$ is generated by 3-cycles given $n\geq 3$. Is this proof correct?

The elements of $A_n$ is either of the form $(a,b,c,...)...$ or of the form $(a,b)(c,d)...$ In both cases, the element is a product of an even number of transpositions, not pairwise disjoint in the ...
0
votes
1answer
7 views

In $\mathbb{Z}_{79}$, $(\alpha\gamma + 66\beta\delta, \alpha\delta + 6\beta\gamma) = (0,0) \implies (\gamma, \delta) = (0,0)$

How does one prove that, in $\mathbb{Z}_{79}$, if $(\alpha, \beta) \neq (0,0)$, then $$(\alpha\gamma + 66\beta\delta, \alpha\delta + 6\beta\gamma) = (0,0) \implies (\gamma, \delta) = (0,0).$$ This ...
1
vote
1answer
23 views

Is every representation of $G$ over $\mathbb{K}$ trivial?

True/False: Let G be a group with $|G| = p^n$ for a prime number $p$ and $n \in \mathbb{N}$, and let $\mathbb{K}$ be a field of characteristic $p$. Is every representation of $G$ over $\mathbb{K}$ ...
2
votes
1answer
35 views

Normalizers of subgroups of Sylow $p$-subgroups

I am wondering whether there is an easy example of a finite group $G$ with a Sylow $p$-subgroup $P$ and a subgroup $Q\leq P$ such that the normalizer $N_P(Q)$ of $Q$ in $P$ is NOT a Sylow $p$-subgroup ...
1
vote
1answer
40 views

Quotient of direct sum of abelian groups

Let $A \oplus B \simeq A' \oplus B $. Does it follow that $A\simeq A'$? Many thanks in advance!
1
vote
0answers
20 views

references of discrete association scheme

I tried to find a book or paper to understanding discrete association scheme but I could not get any book for that. What is the good references for that?
2
votes
2answers
44 views

Show $p(x)$ is a primitive polynomial

First the definition: Polynomial $q(x) \in \mathbb{Z}_p[x]$ of degree $n$ is called primitive, iff: $q(x) \mid x^{p^n-1}-1$ $\forall k : 1 \leq k \leq p^{n}-1$ : $q(x) \nmid x^k - 1$ ...
0
votes
1answer
32 views

Noncommutative rings and the evaluation homomorphism

Recall the evaluation homomorphism of a ring. For example, if $\{R[x]|{p(x)=a_0+a_1x+a_2x^2...}$} is the ring of polynomials with real coefficients then we can evaluate with respect to $c$ by letting ...
2
votes
1answer
49 views

group action same thing as homomorphism

A linear group action of a group $G$ on a vector space $V$ is the same thing as a homomorphism from G to the general linear group $GL(V)$. attempt: Suppose a linear group action of a group $G$ on a ...
2
votes
1answer
21 views

Order of element in polynomial ring in Hatcher

So I've been reading Hatcher and I am unsure what they mean when they say things like $H^\ast(\mathbb{R}P^n;\mathbb{Z}_2)\cong \mathbb{Z}_2[\alpha]/(\alpha^{n+1})$ where $|\alpha|=1$. It is this last ...
0
votes
1answer
73 views

If $f:\mathbb{Z} \to \mathbb{Z}$ is an isomorphism, prove that $f$ is the identity map. [on hold]

I am a little baffled by this question. Is it safe to assume that since $f$ is an isomorphism, $f (1) = 1$ ? And, if it is safe to assume this, could I construct a proof by induction, by using the ...
2
votes
2answers
22 views

Let $\phi: K(X) \rightarrow K(X)$ be the K-morphism such that $\phi (X)=1-X$. Find an element $Y \in K(X)$ so that $K(Y)=\{ f\in K(X) : \phi (f)=f\}$.

Let $K(X)$ be the field of rational functions of $X$ over some field $K$. Let $\phi: K(X) \rightarrow K(X)$ be the K-morphism such that $\phi (X)=1-X$. We have $L:=\{ f\in K(X) : \phi (f)=f\}$. Find ...
1
vote
0answers
23 views

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 ...
0
votes
0answers
21 views

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 $m<t$? ...
1
vote
3answers
53 views

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 + ...
0
votes
0answers
31 views

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 ...
1
vote
0answers
21 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 ...
1
vote
1answer
26 views

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

Let $p$ be a prime number. I've to figure out for which $p$ $$G_p = \{\sigma \in S_{12}: ord(\sigma) \text{ divides } p \}$$ is a subgroup of $S_{12}$. I realize that we have to consider ...
2
votes
1answer
37 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} ...
3
votes
1answer
36 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 ...
0
votes
1answer
31 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 ...
2
votes
3answers
74 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 ...
1
vote
1answer
20 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 ...
1
vote
2answers
17 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 $K \subset K(b)$. ...
-2
votes
0answers
19 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 $R$
1
vote
3answers
40 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 ...
-2
votes
0answers
33 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 ...
3
votes
1answer
61 views

$\Bbb R,\Bbb C,\Bbb H$ 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 ...
1
vote
1answer
31 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 ...
0
votes
1answer
18 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 $y|x$. ...
3
votes
2answers
27 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 ...
0
votes
2answers
33 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 ...
2
votes
2answers
35 views

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 ...
1
vote
1answer
45 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) ...
1
vote
0answers
20 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 ...
2
votes
1answer
50 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, ...
-1
votes
2answers
41 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?
3
votes
2answers
145 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 ...
4
votes
4answers
288 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 = ...
0
votes
0answers
27 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?
0
votes
2answers
25 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 ...
0
votes
1answer
46 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 ...