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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8
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2answers
113 views

Series with $\zeta$

How do I calculate the following series: $$ \zeta(2)+\zeta(3)+\zeta(4)+ \dots + \zeta(2013) + \zeta(2014) $$ All I know is that $\zeta(2)=\pi^2/6$ and $\zeta(4)=\pi^4/90$. But this is not enough to ...
1
vote
1answer
93 views

How many non isomorphic semidirect products are there between $\mathbb Z_2$ and $SL(2,3)$?

I know that $GL(2,3)$ is one of this, but i need the characterization of all possibles of the semidirect products between $\mathbb Z_2$ and $SL(2,3)$. Thanks, for any help.
0
votes
2answers
71 views

The cardinality of elliptic curves over finite field

Given an elliptic curve over $\mathbb Q$ as $y^2=f(x)$ where $f(x)$ is a cubic polynomial. In some places I read that if $p$ is a prime of good reduction then we have that $E(\mathbb F_p)=p+1$. Is ...
3
votes
3answers
78 views

How many elements does this ring have?

I know that the following ring is not a field because the defining polynomial is reducible into two polynomials that are irreducible: $$\mathbb Z_2[X]/(x^5+x+1)$$ where $$x^5 + x + 1 = (x^2 + x + 1) ...
0
votes
1answer
91 views

Why does this imply that $x \in S$?

I am reading the proof of the following proposition at a paper but I got stuck at some points... Proposition 1. Let $R$ be an integral domain of characteristic zero. Suppose there exists a subset ...
1
vote
0answers
33 views

Existence of a splitting ring

Let $R$ be a commutative ring and $f\in R[X]$ be a monic non-constant polynomial. How can one show that there exists a commutative ring $S$ so that $R$ is a subring of $S$ and $f$ can be written as a ...
4
votes
1answer
56 views

Show that $1\otimes (1,1,\ldots)\neq 0$ in $\mathbb{Q} \otimes_{\mathbb{Z}} \prod_{n=2}^{\infty} (\mathbb{Z}/n \mathbb{Z})$.

Show that $1\otimes (1,1,\ldots)\neq 0$ in $\mathbb{Q} \otimes_{\mathbb{Z}} \prod_{n=2}^{\infty} (\mathbb{Z}/n \mathbb{Z})$. Here's what I tried: If $1\otimes (1,1,\ldots)= 0$, then $1\otimes ...
20
votes
5answers
3k views

Proving that all integers are even or odd [duplicate]

I know that $\mathbb{Z}$ is a group under addition with a multiplication defined. I have just the definition of even and odd integers: $n$ is even if $n = 2k$ for some integer $k$ and $n$ is odd if $n ...
0
votes
1answer
42 views

Let $ M $ be the smallest normal subgroup of $ K $ such that $ K/M $ is nilpotent. What's mean smallest normal subgroup?

theorem: Let $ G $ be solvable with $ \Phi(G)=1 $ and assume that each minimal normal subgroup has prime order or order $ 4 $. Then every chief factor of $ G $ has prime order or is $ G $-isomorphic ...
-2
votes
1answer
28 views

how many elements are there which has order 3?

if the number of sylow-3 subgroups of a group of order 96 is 4, how many element are there which has order 3? I dont know how to start. why isnt the answer 4?
0
votes
1answer
39 views

Smallest normal subgroup and minimal normal subgroup, what's the difference?

Let $ G $ is a finite group and $ N $ be a minimal normal subgroup of $ G $ and $ M $ be a smallest normal subgroup of $ G $. Smallest normal subgroup and minimal normal subgroup, what's the ...
1
vote
2answers
58 views

which one of these may not be abelian?

If a group G has these orders. which one of these may not be abelian? 4,31,55,39 and 121 since 4 and 121 is prime square. they are abelian. and 31 is prime therefore cyclic so abelian. what about 55 ...
1
vote
0answers
46 views

Countably many projections on more than continuos vector space with trivial commutant?

Is there such an example? An $\mathbb{F}_2$-vector space $V$ of dimension strictly more than the continuos $c=|2^{\mathbb{N}}|$, and a numerable set of commuting $\mathbb{F}_2$ projections ...
0
votes
0answers
21 views

Relation between simple and indecomposable modules

I know that every simple module is indecomposable but the reverse case isn't true in general. So is there a way or conditions that we can add on the indecomposable modules so that they are a simply ...
1
vote
1answer
35 views

Functor $M\otimes_B -$ is left and right adjoint to $\operatorname{Hom}_A(M,-)$ when there is a symmetrizing form for $A$?

Suppose $A$ is an $R$-algebra over a commutative ring $R$, which is finitely generated and projective as an $R$-module. A symmetrizing form is a map $t\in\operatorname{Hom}_R(A,R)$ such that ...
3
votes
1answer
46 views

Adjoining an identity to a ring

I am run into the following in an Algebra text: "Let $R_0=\mathbb Z/2\mathbb Z⊕\mathbb Z/2\mathbb Z⊕\cdots$ viewed as a ring without identity, with addition and multiplication defined componentwise. ...
-4
votes
1answer
27 views

Binary word addition; error pattern

If a word $a = a_1,a_2,...,a_n$ is sent is sent (this is in regards to coding/IT/etc.-I'm trying not to include any extraneous information) and a word $b= b_1,b_2,...,b_n$ is received (where the ...
5
votes
2answers
56 views

Show that if $|G| = 30$, then $G$ has normal 3-Sylow and 5-Sylow subgroups.

Show that if $|G| = 30$, then $G$ has normal $3$-Sylow and $5$-Sylow subgroups. Let $n_3$ denote the number of 3-Sylow subgroups and $n_5$ the number of $5$-Sylow subgroups. Then, by the third ...
0
votes
1answer
53 views

Dilemma with the classification theorem of finite groups

We know that if $H < G$ , $G$ commutative, and $G/H \cong \hat H < G$, then $ G \cong H \oplus \hat H$. Then on the basis of this I could write $Z_4 \cong Z_2 \oplus Z_2 $ but we know that $Z_4 ...
2
votes
2answers
62 views

What explicitly is the “adjunction” isomorphism $Hom_C(A,C)\to Hom_B(A,Hom_C(B,C))$?

Suppose $B$ and $C$ are commutative rings, $A$ a $B$-algebra, and $B$ is a $C$-module. What exactly is the "adjunction" isomorphism $$Hom_C(A,C)\to Hom_B(A,Hom_C(B,C))?$$ Given $A\to C$, it needs to ...
1
vote
1answer
79 views

Fundamental Theorem of Abelian Groups

From fundamental theorem of finite abelian groups I can say any finite abelian group $G$ is isomorphic to direct sum of cyclic groups i.e, $G\cong Z_{{p_1}^{i_1}}\oplus Z_{{p_2}^{i_2}}\oplus ...
0
votes
0answers
33 views

Maximal ideal of polynomial ring over a subfield

Let $L/K$ be an algebraic extension of fields. Let $B = L[X,Y]$ and $A = K[X,Y]$. Suppose $a$, $b \in L$ and $m = (X-a,Y-b)$ is an ideal of $B$. Show that $m$ and $m \cap A$ are maximal ideals of ...
3
votes
0answers
58 views

Reference Request: Group Theory via the Group Action Perspective

I am looking for a higher undergraduate or graduate level textbook that introduces group actions after groups just as many textbooks introduce modules after rings. I think the semigroup/semigroup ...
2
votes
0answers
29 views

p-group of class 2 with cyclic commutator subgroup

I was reading a paper "ON THE EXISTENCE OF NONINNER AUTOMORPHISMS OF ORDER TWO IN FINITE 2-GROUPS" you can get it from here. In Introduction page 2 He said "It is worth mentioning here that we need ...
1
vote
0answers
27 views

$ G = MC $ for some cyclic subgroup $ C $. If $ \vert G : M \vert = 4 $ then why $ G \cong S_{4} $?

Let $ M $ is a maximal subgroup of finite group $ G $, that $ G = MC $ for some cyclic subgroup $ C $. If $ \vert G : M \vert = 4 $ and $ M_{G} = 1 $ then why $ G\cong S_{4} $?
1
vote
0answers
29 views

proof that addition multiplication and < are well defined on the Hyperreals

Could someone please check my proof for showing addition, multiplication and < are well defined on the Hyperreal numbers. Proof: Suppose $$[a_n]=[b_n]$$ and $$[c_n]=[d_n]$$ where [] denotes the ...
-2
votes
1answer
69 views

Will this punch a hole in the field of complex number? [closed]

According to this, complex number is algebraically closed, i.e. every polynomial has complex root. What if we allow other type of equations? I ask this question because equations seemingly can extent ...
2
votes
2answers
201 views

A group homomorphism from a simple group is injective

Let $G_1$ be a simple group, that is the only normal subgroups of $G_1$ are itself and the trivial subgroup. If $\phi : G_1 \rightarrow G_2$ is a group homomorphism, does that mean $\phi$ is ...
0
votes
1answer
52 views

Finding the eigenvalue and eigenspace for a function defined as the intersection of element of A and B [closed]

Let $S$ be a non empty set and let $V_S$ be the set of all the subsets of $S$. $V_S=\{A:A⊂S\}$ Now fix an element $B$ of $V_S$ and define a function $f:V_S \rightarrow V_S$ by $f(A)=A∩B$. Find the ...
0
votes
1answer
56 views

Every minimal normal subgroup is contained in the center

G is a finite group in the following questions: (X):Every minimal normal subgroup is contained in the center. (1) Let $N$ and $M$ be normal subgroups of $G$, both of which satisfy (X), then prove: ...
0
votes
1answer
20 views

Algorithmic solultion for eigenproblem over finite field

i am looking for the standard algorithms for solving eigenvalue problems over finite fields. (For example the algorithm implemented in GAP). I googled a lot but did not come to a conclusion. I saw ...
2
votes
2answers
52 views

Can the image of a set under a permutation be a proper subset of the set itself?

I'm working on some exercise in Fraleigh's "A First Course in Abstract Algebra" and one of them involves permutations under which the image of a certain set is a subset (proper or improper) of the set ...
2
votes
1answer
53 views

Scalar product on Lie algebra of compact Lie group [duplicate]

I am studying Differential Geometry and I am facing with a lemma in which there is a step that I do not understand. In particular, let $G$ be a connected compact Lie group, is used "$\langle\ \cdot , ...
0
votes
1answer
34 views

Why is $Y_n$ of that form?

Let $R$ be any integral domain of characteristic zero. We consider the Pell equation $$X^2-(T^2-1)Y^2=1\tag 1$$ over $R[T]$. Let $U$ be an element in the algebraic closure of $R[T]$ satisfying ...
-2
votes
1answer
31 views

Number of subfields of a given field [duplicate]

Let $F$ be a field with $5^{12}$ elements. What is the total number of proper subfields of $F$? A) $3$ B) $6$ C) $8$ D) $5$ Explain the concept used to solve the question.
1
vote
1answer
20 views

Order of multypling 2 sub-groups who's orders are coprime

Im given as an exercize to prove that an order of 2 sub-groups A,B who's orders are coprime, is: $$|A| \cdot |B|$$ What I know that generally: $$|AB|=\frac{\left|A\right|\cdot ...
0
votes
1answer
22 views

Identity element of word addition

I realize this is rather an arbitrary question, but it's important to me, that I understand it and get it right, and I'm not finding the answer anywhere else. I'm working through "A Book of Abstract ...
1
vote
1answer
12 views

conditions on multypling sub-groups (so it would be a new sub-group)

Ok, so we know that if G is abelian and A,B are here sub-groups, then AB, defined by: $AB\:=\:\left\{ab\::\:a\in A,\:b\in B\right\}$ is a new sub-group. Now, I'm given another condition and I need ...
0
votes
3answers
40 views

Verify that Function is Homomorphism for $\mathbb{Z}_8 \rightarrow \mathbb{Z}_4$

Prof. Pinter's "A Book of Abstract Algebra" presents the following exercise: Consider the function $f: \mathbb{Z}_8 \rightarrow \mathbb{Z}_4$ given by: $f = (0 \rightarrow 0, 1 \rightarrow 1, ...
5
votes
1answer
72 views

If $|G|= 2^{3}3^{3}11$, then $G$ is not simple

I have this problem in my notes: If $|G|=2^{3}3^{3}11$, then $G$ is not simple The instructor solved it in a way that I could not follow. The solution I have is attached below. If someone could ...
0
votes
0answers
34 views

A Condition for Normality of a Subgroup involving Bijections

Let $G$ be a group, and let $H \leq G$. Suppose the map $\phi: Hx \rightarrow xH$ is a bijection. Now, in general, such a map is not even well-defined. I trying to use this fact to show that $xH=Hx ...
3
votes
1answer
52 views

Does $f\otimes_A 1_{A/m}:M\otimes A/m\to N\otimes A/m$ injective for all maximal $m$ imply $f$ is an isomorphism?

Let $A$ be a commutative ring. Suppose $f\colon M\to N$ is a morphism of free $A$-modules of equal, finite rank. If $f\otimes_A 1_{A/m}$ is injective for all maximal ideals of $A$, does this imply ...
1
vote
0answers
34 views

Making sense of multiplication in tensor product

Let $\mathbb{Z} / p$ and $\mathbb{Z} / q$ be $\mathbb{Z} /pq$ modules where $p$ and $q$ are distinct primes. Then $\mathbb{Z}/p \times \mathbb{Z}/q$ is a $\mathbb{Z} / pq$ module Now let $$ i : ...
2
votes
2answers
194 views

Why is there a $p\in \mathbb{N}$ such that $mr - p < \frac{1}{10}$?

I am reading the following part of the paper of Denef : Let $R$ be a commutative ring with unity and let $D(x_1,\dots , x_n)$ be a relation in $R$. We say that $D (x_1,\dots , x_n)$ is diophantine ...
3
votes
1answer
81 views

Splitting field of $x^4+3$

Let $K$ be the splitting field the polynomial $x^{4}+3$ over $\mathbb{Q}$.Find the Galois group of $K$ over $\mathbb{Q}$? I think $[K:\mathbb{Q}]=8$ but how can we find the group? Any help would be ...
0
votes
1answer
23 views

A direct sum of cyclic modules over a PID

Let $R$ be a PID, let $p$ be a prime of $R$, and let $M$ be the $R$-module $R/Rp^{e_1}\oplus \cdots \oplus R/Rp^{e_n}$ where the $e_i$ and $n$ are positive integers. Define $M(p)=\{m: pm=0\}$ and ...
0
votes
0answers
17 views

If $M$ is an $A$-module via $\varphi\colon A\to\operatorname{End}(M)$, $\mathrm{coker}(\varphi)\otimes M$ is projective?

I have a brief passage I don't understand. Suppose $R$ is a commutative ring, $A$ is an $R$-algebra, which is projective and finitely generated as an $R$-module. Let $M$ be a progenerator for $A$, so ...
1
vote
1answer
47 views

How to find the kernel of $\sigma: \Bbb Z_{17}\to G $

Let G be a group,$ e_G$ is the unit of G and $\sigma: \Bbb Z_{17}\to G $ is a homomorphism which is not injective. so find $Ker(\sigma)$ it is probably wrong but we know $Ker(\sigma)\neq\{\bar ...
3
votes
1answer
24 views

Let $A$ be a finitely generated $R$-module. Show that if $A/MA$ can be generated by $n$ elements then so can $A$.

Let $R$ be a local commutative ring with the maximal ideal $M$. Let $A$ be a finitely generated $R$-module. Show that if $A/MA$ can be generated by $n$ elements then so can $A$. I tried to apply ...
1
vote
0answers
38 views

Are all elements of groups Garside elements?

Here in the definition of Garside Element, it seems that if the monoid here is a group, then all its elements are Garside elements simply because all elements are both left and right divisors of any ...