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
1answer
9 views

Cartesian product to direct sum

I have no idea, how to prove rigorously the corollary from the proposition. I know that i can use the isomorphism $\phi:x_1e_1+...+x_me_m \in \oplus_i^mvect(e_i)\to (x_1e_1,...,x_m e_m) \in ...
0
votes
1answer
33 views

Show that no ring containing R can contain a root of g(x) = 3x +1

Show that if $R = \mathbb Z_6$ and $g(x) = 3x + 1 ∈ R[x]$, then $R[x]/(g(x)R[x])$ does not contain a root of $g(x)$. More generally, show that no ring containing $R$ can contain a root of $g(x)$. ...
0
votes
2answers
32 views

Field characteristic for a finite product of fields of characteristic $0$

Kind of a silly question, but is a finite product of fields of characteristic $0$ also of characteristic $0$? For instance, $\mathbb{C}$ has characteristic $0$, but then does $\mathbb{C}^n, n>1$ ...
1
vote
2answers
35 views

Ring of matrices has no nontrivial ideals [duplicate]

It is a theorem that a commutative ring is a field if and only if it has no nontrivial ideals. Clearly this does not hold in the noncommutative case. I am trying to show for instance that the ring of ...
0
votes
0answers
24 views

Linear Algebra. Is this question realte to combination and factorials?

I am not able to understand this question and what is the entries of matrix A exactly. Question Thanks.
0
votes
0answers
34 views

Reducibility over $Q$ implies reducibility over $Z$.

Let $f(x) \in$ $Z[x]$ , if $f(x)$ is reducible over $Q$ , then it is reducible over $Z$. I went through the proof from the book I'm reading , which starts as follows : We're given $f(x)$ is ...
1
vote
1answer
26 views

Two non-isomorphic groups which are epimorphic images of each other

Let $G$ and $H$ be two groups, I am looking for an example such that $G$ is an epimorphic image of $H$ and $H$ is an epimorphic image of $G$ (i.e. they are both quotients of the other group), but they ...
2
votes
1answer
21 views

Group of exponent $2$.

When I have a group $G$ of exponent $2$ and I know that all elements in $G-\{e\}$ are conjugated, is it right that $G$ is of order $2$? My try: For $g,h \in G - \{e\}$ the conjugation assumption ...
0
votes
1answer
18 views

A $R$-module over a ring $R$ with identity is free iff it is a direct sum of copies of $R_R$, where $R_R$ is $R$ considered as a module over itself

Let $R$ be a ring with identity and $M$ be an $R$-right module. Then $M$ is called free over $X \subseteq M$ if for every module $N$ with mapping $\alpha : X \to N$ we can extend it uniquely to a ...
1
vote
2answers
39 views

Group of finite order where every element has infinite order

An often given example of a group of infinite order where every element has infinite order is the group $\dfrac{\mathbb{(Q, +)}}{(\mathbb{Z, +})}$. But I don't see why every element necessarily has ...
0
votes
0answers
20 views

Subgroup of square generators. [duplicate]

Let $N$ be a subgroup of $G$ with $N$ being generated by $\{x^2|x \in G\}$. Prove that $N$ is a normal subgroup of $G$. And that $[G, G] \subset N$. My idea was to look at $G/N$ and take $f:G ...
8
votes
1answer
92 views

What's an Isomorphism?

I'm familiar with the definition (inverses and bijections, preserving operations) in the context of groups and vector spaces, the hoeomorphism of topological spaces, and have some feeling for the ...
0
votes
1answer
28 views

Is/when is this property about the totative subset of $(\mathbb{Z}_n , +_n)$ true?

Is it possible to show (or when is it true); that for the group $\mathbb{Z}^+_n :=(\mathbb{Z}_n , +_n) $, there exists an $a \in \mathbb{Z}^+_n$ for each $z \in \mathbb{Z}^+_n$, where both $z+_n a$ ...
1
vote
0answers
25 views

Criterion for isomorphism of two groups given by generators and relations

When are two presentations of groups are isomorphic? In this post it is said: [...] find a set of generators of the first group that satisfies the relations of the second group [...] But I doubt ...
1
vote
0answers
20 views

Compact abelian group and Continuous functions

If $G$ is a compact abelian group, $\widehat{G}$ is the dual group of $G$,i.e. all the continuous homomorphism from $G$ to $S^1$,$S^1=\{z\in \mathbb{C}\big | |z|=1\}$. Show that the linear span of ...
2
votes
1answer
40 views

Centralizer, normalizer, and center of a dihedral group

Let $A := \{1, r, r^2,..., r^{n-1}\}$. Compute $C_{D_{2n}}(A), N_{D_{2n}}(A),$ and $Z(D_{2n})$. So far I figured that all of the rotations are in the centralizer/normalizer, because all rotations ...
0
votes
0answers
18 views

Set of all permutations on n generating function [duplicate]

Show that $S_n = \langle (1\ 2), (1\ 2\ \ldots\ n) \rangle$ for all $n \geq 2$. I'm not sure how to approach this one.
2
votes
2answers
35 views

Prove that $\mu:G\times G \rightarrow G$ is a homomorphism if and only if $G$ is abelian.

Given $\mu:G\times G$ be the operation on a group $G$; that is, $\mu (a,b)=ab$. Prove that $\mu$ is a homomorphism if and only if $G$ is abelian. I have no problem on proving the necessary ...
0
votes
1answer
20 views

Show that $m\mathbb{Z}/md\mathbb{Z} \cong \mathbb{Z}/d\mathbb{Z}$ if and only if $(d,m)=1$

Show that $m\mathbb{Z}/md\mathbb{Z} \cong \mathbb{Z}/d\mathbb{Z}$ as rings without identity if and only if $(d,m)=1$. I know that if $\phi : A \to B$ is a epimorphism ring and $A$ is a unit ...
0
votes
1answer
17 views

Compute the matrix with the standard basis.

Let $X$ be the left representation of $S_3$. Compute the matrix $X(132)$ in the standard basis $S = {\{123,132,213,231,312,321}\}$. attempt: In general if $G = S_3 = {\{\pi_1, \pi_2,..,\pi_6}\}$, ...
0
votes
4answers
33 views

Show that $\ker(\phi)$ is a maximal ideal if and only if $B$ is a field

Let $A$ and $B$ two commutative rings with unity $1_A \not= 0_A$ and $1_B \not= 0_B$. Consider $\phi : A \to B$ a ring epimorphism. Show that if $\ker(\phi)$ is a maximal ideal, $B$ is a field. I ...
1
vote
1answer
15 views

Let $n \geq 2$ be an integer, and consider the group $Z_n:=(\{0,1,. . .,n-1\}, +_n)$. Let $k \in Z_n$ \ $\{0\}$.

Let $n \geq 2$ be an integer, and consider the group $Z_n:=(\{0,1,. . .,n-1\}, +_n)$. Let $k \in Z_n$ \ $\{0\}$. Show that the following statements are equivalent: (a) $\gcd(n,k)=1$, (b) the only ...
0
votes
0answers
16 views

Simplifying a coset

Let G be a group and let $M,N \leq G$ be normal such that $G = MN$. Prove that $G/(M \cap N) \cong (G/M) \times (G/N)$ I have found a solution to this question here: ...
1
vote
1answer
29 views

First isomorphism theorem application

Let G be a group with, $N\subset G$ a normal subgroup, And assume that $H$ is a subgroup of $G$, $H\subset G$. Further $HN=G$ and $H\cap N = \{e\}$ . Prove that $H$ generates the cosets of $N$ in ...
-2
votes
0answers
14 views

How to find identity element of a set (under modular) operation?

Question 1) Can the set of $\{0, 1, 2, 3\}$ under the operation of modulo-$4$ addition and multiplication form a group as well as a field ? If yes then how and if not then why ? Question 2) How to ...
0
votes
0answers
16 views

Regular semigroups- normal semigroups!

Can someone please help me to understand this : If $S$ is a Clifford semigroup with the set of idempotents $E$, then $S'$ be a sub-semigroup of $S$ ( so $S$ be a semilattice with the same set of ...
0
votes
0answers
34 views

Unramified morphism

I was reading the following page: https://ayoucis.wordpress.com/2014/04/06/unramified-morphisms/ and there are several things I do not understand and would like to clarify. First doubt The ...
2
votes
1answer
31 views

Can this extension of fields be transcendental?

Let $(R, \mathfrak m)$ be a local integral domain which is contained in a field $K$. Let $0 \neq x \in K$ be such that $\mathfrak m R[x]$ is a proper ideal of $R[x]$ (one can show for any $x$ that ...
2
votes
1answer
24 views

Finite abelian group as Z-module

If $M$ is a finite abelian group then $M$ is naturally a $Z$-module. Can this action be extended to make $M$ into a $Q$-module ?
1
vote
1answer
22 views

What does it mean a transitive permutation?

let $X$ be a finite set. Let G be a group. What is the meaning of $G$ is a transitive permutation on set $X$?
1
vote
0answers
32 views

Group theory: Intuition as to what a group is [duplicate]

In group theory the group is an algebraic structure consisting of a set which has elements associated with definite finitiary operations. Can an intuitive explanation be provided as to what this ...
0
votes
0answers
10 views

Representation of left ideals of the matrix ring [duplicate]

We know that $J$ is an ideal of the full matrix ring $S=M_n(R)$ if and only if $J$ is the ring of all $n\times n$ matrices over $I$ for some ideal $I$ of the ring $R$ with identity. Now, my question ...
-1
votes
1answer
34 views

Doubt about associative property of a group (Abstract Algebra). [on hold]

I am new to abstract algebra and I have a doubt about the associative property. Suppose a set is given, such as $G=\{0,1,2,3,4\}$ under $\pmod{5}$ addition operation and we have to check whether $G$ ...
0
votes
1answer
26 views

Prove $(H \times 1)(1\times K)= H\times K$ where $H,K$ are groups.

Prove $(H \times 1)(1\times K)= H\times K$ where $H,K$ are groups. Suppose $x=ab,a\in H\times 1,b\in 1\times K$ Then $x=(h,1)(1,k)$ where $h\in H,k\in K$ Hence $x=(h,k)\in H\times K$ Let ...
1
vote
1answer
24 views

Classifying groups of order $6$ using semidirect products

Let G be a group of order 6. I am able to do the exercise without semidirect products($G \cong Z_6 $ or $S_3$) but I don't know how to use semidirect products to do this. By Sylow's theorem, there ...
4
votes
1answer
47 views

Why do polynomials and integers both have a long division algorithm?

The grade-school long division algorithm and the polynomial long division algorithm are identical, if I'm not mistaken. Why is this the case? Are the two algebraic structures identical in some sense? ...
1
vote
0answers
53 views

Spliting subspaces and fields

I'm sure that the following is true, but I can't prove it. Let $R<S<K, R=\mathrm{GF}(q),\ S= \mathrm{GF}(q^n), \ K= \mathrm{GF}(q^{mn})$ be a tower of finite fields and $A = \{\theta\in K: ...
1
vote
2answers
40 views

$2\otimes 1$ is non-zero in $2\mathbb Z\otimes_{\mathbb Z} \mathbb Z/2\mathbb Z$.

I had the following doubt: Show that the element $2\otimes 1$ is $0$ in $\mathbb Z\otimes_{\mathbb Z} \mathbb Z/2\mathbb Z$ but not a zero in $2\mathbb Z\otimes_{\mathbb Z} \mathbb Z/2\mathbb Z$. ...
0
votes
3answers
30 views

Prove that the four roots of unity form an abelian multiplicative group

My question is: Let $i = \sqrt{-1}$. Prove that the four roots of unity $\{1, -1, i, -i\}$ form an abelian multiplicative group. I know that abelian group is a group with commutative property. ...
0
votes
1answer
32 views

Order of a subgroup generated by two permutations

Let $$\alpha=(1,3,12,7)(8,5,6,2,11)(4,9,10)$$ $$\beta = (1,5)(6,8,11)(12,3,2,7)(4,9,10).$$ How does one prove that a subgroup G that contains $\alpha$ and $\beta$ has order $o \ge 120$? ...
-1
votes
0answers
34 views

Permutations of $S_n$ whose order divides a positive integer $m$

For which $n,m \in \mathbb{N}$ is $$K_m = \{\sigma \in S_{n}: \text{ord}(\sigma) \text{ divides } m \}$$ a subgroup of $S_{n}$? How does one approach such a problem?
1
vote
2answers
32 views

Adjoint pair of functors and cogenerator elements

Let $F:\mathcal{A} \rightarrow \mathcal{B}$ and $G:\mathcal{B} \rightarrow \mathcal{A}$ be additive functors between abelian categories, such that $(F,G)$ is an adjoint pair. If $B \in \mathcal{B}$ is ...
0
votes
2answers
41 views

Is $(\mathbb{Z}_4, +_4)$ isomorphic to $(\langle i\rangle, *)$

Now, I am not sure, but I think that the second group is cyclic, because of the way it's defined $(\langle i\rangle,*)$. $i$ is probably the generator of the group. But, how can I prove that ...
1
vote
1answer
27 views

A group $G$ is finitely generated iff if there is a surjective homomorphism $F(\{1,…,n \}) \to G$

This is taken for granted in Algebra: Chapter 0 by Paolo Aluffi. Here is a definition of subgroup generated by a subset from the book: Let $A \subseteq G$. We have a ujnique group homomorphism ...
1
vote
1answer
50 views

Find error in abstract algebra proof

I suspect that the proof below is flawed. I did not use the hypothesis "$\ker(h) \subseteq \ker(k)$" when proving sufficiency. Lemma. $ $ Let $G$, $H$, $K$ be groups, let $h : G \to H$ and $k : G ...
0
votes
1answer
33 views

Suppose that $f : C → C$ is an isometry such that $f(0) = 0$, $f(1) = 1$ and $f(i) = −i$. Prove that $f(z) = \bar z$ for all $z ∈ C$.

Suppose that $f : C → C$ is an isometry such that $f(0) = 0$, $f(1) = 1$ and $f(i) = −i$. Prove that $f(z) = \bar z$ for all $z ∈ C$. I already have a proof for this but I would like an explanation ...
1
vote
1answer
29 views

Describing the clopen sets of a profinite group

I've read somewhere that all clopen subsets of a profinite group $$G \simeq \varprojlim\left(G_i, f_{ij}:G_i \to G_j\right)_{i,j \in I}$$ are exactly the preimages of subsets of the $G_i$'s. It's easy ...
0
votes
2answers
17 views

Show that the set of all principal ideals is an equivalence class of the relation $\sim$

Let $A$ a integral domain and let $\mho(A)$ the set of all non-zero ideals. Show that the set of all principal ideals is an equivalence class of the relation $\sim$ that we can noted by $[A]$. ...
4
votes
1answer
55 views

Norm on “tower” of fields (the question comes from algebraic number theory)

Consider the field extensions $L\supset K'\supset K$, where both $L|K$ and $K'|K$ are finite and Galois. I want to prove that $$N_{L|K}(L^\ast)\subseteq N_{L|K'}(L^{\ast})$$ Maybe it is very easy, ...
1
vote
1answer
27 views

What are the $GL_n(F)$-orbits of a group action on the set of idempotent matrices?

Let $S= \{A \in M_{n \times n}(F):A^2=A\}$ (set of idempotent matrices). The general linear group $G=GL_n(F)$ acts on $S$ by $A.g=gAg^{-1}$ (conjugation). I'm having trouble visualizing the ...