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

learn more… | top users | synonyms (1)

0
votes
1answer
23 views

Correctness of proof that the commutative operation * on a binary structure is a structural property [duplicate]

Here is what I have as a proof for now. Can you tell me where I need to edit it and or how it should be instead? Let and be two arbitrary binary structures with an isomorphism f: S->T. Assume * is ...
1
vote
2answers
51 views

what is the order of the subgroup of $S_3$

True or false? $S_{3}$ has a subgroup of order $5$. $S_3$ is the group of all permutations of the numbers $1,2,$ and $3$. An example of a permutation is, for instance, $(32)$, which means switching ...
-2
votes
0answers
29 views

Composition series in an alternating group

I have seen that a composition series for $A_4$ is $$1<\{1, (1 2)(3 4)\}<\{1,(1 2)(3 4),(1 3)(2 4),(1 4)(2 3)\}<A_4$$ But how can I find it? I mean when I am asked to find a composition ...
-4
votes
1answer
49 views

mapping $S_4$ a nd checking if it is cyclic [duplicate]

True or false? Every element of $S_4$ is a cycle. Can anyone help me in how to solve this question? I find difficulty in answering the question.
0
votes
2answers
73 views

What does “finding an element” in $\mathbb Z_n$ mean?

I am currently in my last year of high-school and I try to learn Algebra on my own, one of my textbook exercise ask me to Find elements: $$ ...
1
vote
0answers
51 views

Is the inverse limit of a Mittag-Leffler system of finitely generated modules countably generated?

Let $R$ be a ring and let $\{M_n,\varphi_n\}_{n\in \mathbb{N}}$ be a projective system of finitely generated modules which satisfies the Mittag-Leffler condition. Is it true that $\varprojlim M_n$ is ...
1
vote
1answer
16 views

How to show that any solvable transitive subgroup of S$_p$ where $p$ is a prime has a conjugate contained in Aff($\mathbf F_p$)?

Here Aff ($\mathbf F_p$) denotes the group of affine transformations $x\rightarrow ax+b,$ with $ a\neq 0, b\in \mathbf F_p$. What I've done is to show that the penultimate group in the solvable series ...
1
vote
2answers
59 views

Is this binary operation a group?

Let $Y=\left\{(a,b)\in\mathbb{R}\times\mathbb{R}\ |\ a\ne 0\right\}$. Given $(a,b),(c,d)\in Y$, define $(a,b)∗(c,d)=(ac,ad+b)$. Prove that $Y$ is a group with the operation $*$. I already did the ...
1
vote
1answer
42 views

Prove that an associative operation* is a structural property

First off how does one prove that an operation $\ast$ is a structural property and how is this different from just proving an isomorphism? My main question is: how can I prove that $\ast$ on a binary ...
6
votes
1answer
94 views

factoring $x^n+x+1$

Is there a way of factoring a polynomial of the general form $$x^n+x+1$$ in the ring $\mathbb C[x]$ or $\mathbb R[x]$ or $\mathbb Z [x]$ for any $n \in \mathbb N$? (Or perhaps with certain conditions ...
-1
votes
2answers
58 views

How do you calculate this product $\mathbb{Z}_6\times\mathbb{Z}_6$

How do direct product (Cartesian product) of $$\mathbb{Z}_6\times\mathbb{Z}_6$$ I need to know if this product is a direct integral domain (ring integrity).
1
vote
1answer
44 views

How many irreductible polynoms of degree $n=3$ are there over $\mathbb Z_{3}=\{\overline{0}, \overline{1}, \overline{2}\}$?

How many irreductible polynoms of degree $n=3$ are there over $\mathbb Z_{3}=\{\overline{0}, \overline{1}, \overline{2}\}$? Please, check my solution: $f(x)$ is a polynom of degree $n=3 (\delta f=3) ...
0
votes
2answers
43 views

Ring homomorphism from $\Bbb Q$ into a ring

Let $A$ be a ring. I'm trying to prove that there is only one ring homomorphism (different from the zero one) from $\Bbb Q$ into $A$ or there are no ring homomorphism between $\Bbb Q$ and $A$. I have ...
0
votes
0answers
22 views

Weyl's construction for symplectic groups--an exercise in Fulton and Harris's book

This is an exercise in section 17.3 in Fulton and Harris's book:Representation theory-a first course. Let $V=\mathbb{C}^{2n}$ and $Sp(2n)$ be the symplectic group w.r.t the nondegenerate bilinear ...
-2
votes
2answers
32 views

how to prove this group of the binary operation

Let $Y=\{(a,b)∈ \Bbb R\times \Bbb R∣ a≠0\}$. Given $(a,b),(c,d)\in Y$, define $(a,b)*(c,d)=(ac,ad+b)$. Prove that $Y$ is a group with the operation $*$. I already do the proof of ∗ is an operation on ...
3
votes
1answer
27 views

Constructing a surjection from fundamental group of a mapping cone to Hawaiian Earring to $\prod_\infty \mathbb{Z} / \oplus_\infty \mathbb{Z}$

If X is the subspace of $\mathbb{R}$ consisting of 1, 1/2, ... together with its limit point 0, C is the mapping cone of the quotient map $SX \rightarrow \sum X = $ (the Hawaiian Earring) which ...
5
votes
0answers
105 views

Have these (extremely simple) classes of algebraic structures been considered in the literature? If so, what are they called?

Questions. Have the following kinds algebraic structures been considered in the abstract algebra literature etc.? If so, what are they really called? (I have used made-up terminology for the sake ...
1
vote
1answer
34 views

Prove that there exists a sequence of intermediate fields.

How can I prove that for $K\subset L$ - the Galois extension of degree $p^n$, where $p$ is prime, there exists a sequence $K=K_{0}\subset K_{1}\subset \cdots \subset K_{n}=L$ such that ...
7
votes
1answer
44 views

Are the identities $\{(xy)^p = x^p y^p : p \mbox{ is prime}\}$ logically independent?

For each positive integer $n$, let $\eta_n$ denote the following identity in the language of monoids. $$(xy)^n = x^n y^n$$ For example, $\eta_2$ is the identity $xyxy = xxyy.$ Question. Is it ...
0
votes
1answer
29 views

Show that $\overline{x}\in\Bbb Z/(n\Bbb Z)$ is invertible iff $\gcd(x,n)=1$ [duplicate]

Be $n$ greater than 2 integer and is $\bar x\in\mathbb{Z}_n-\{\bar0,\bar1,\bar2,\ldots,\overline{n-1}\},$ $\;0\leq x<n$. Show that $\exists\; \bar y\in\mathbb{Z}_n$, such that $\bar x\cdot \bar ...
2
votes
0answers
54 views

Connected points problem for $\mathbb Q$

Two points in $\mathbb Q$ are called connected, if their euclidean distance is exactly $1$. You are now allowed to also jump from any point in $\mathbb Q^n$ to another, if they are connected in ...
3
votes
2answers
65 views

finding a map s.t. $\mathbb Q \times C_2$ $\cong$ $\mathbb Q^*$

This is a question from my group theory exam which I was unable to prove: It say that Show that $\mathbb Q \times C_2$ $\cong$ $\mathbb Q^*$ by specifying an isomorphism. But I couldn't find one. ...
2
votes
1answer
64 views

Why is ring of integers $\mathcal O_K$ called ring of integers - what properties of $\mathbb{Z}$ does it inherit?

I was wondering why ring of integers $\mathcal O_K$ for field $K$ is called ring of integers. Definition says that elements in this ring will be a solution for monic equation with coefficients ...
0
votes
1answer
36 views

Proof of $0\rightarrow A^G\stackrel{f}\rightarrow B^G\stackrel{h}\rightarrow C^G\rightarrow 0$

When we have a $G$-modules exact sequence $0\rightarrow A\stackrel{f}\rightarrow B\stackrel{h}\rightarrow C\rightarrow 0$ we have a $G$-modules exact sequence $0\rightarrow ...
1
vote
2answers
84 views

When can a homomorphism be determined entirely by its generators

I read a text which says that: Just because a homomorphism $ϕ :G → H$ is determined by the image of its generators does not mean that any such image will work. e.g.: Suppose we try to define ...
1
vote
1answer
55 views

Operations in $\mathbb{F}_{32}$

I've some difficulties about sums in the field $\mathbb{F}_{32}$. In particular I'm studying an example of a cryptographic attack, where there are a lot of sums in this field, which I don't ...
-1
votes
2answers
56 views

Geometric interpretation of $\mathbb{R}\times\mathbb{R}/H$ for these $H$

$\mathbb{R}\times\mathbb{R}/H$ with: ($G=\mathbb{R}\times\mathbb{R}$) $H=\{(x,0)|x\in\mathbb{R}\}$ $H=\{(x,y)|y=-x\}$ $H=\{(x,y)|y=2x\}$ This is from a book (Pinter's "A book of abstract algebra") ...
1
vote
0answers
29 views

Roots of polynomials modulo prime powers

I'm reading the paper Dearden, Bruce(1-ND); Metzger, Jerry, Roots of polynomials modulo prime powers. European J. Combin. 18 (1997), no. 6, 601–606. link: ...
-1
votes
1answer
51 views

Group Property of element [duplicate]

True or false? If $G$ is a group with the property that $g=g^{-1}$ for all g element $G$, then $G$ is abelian .prove true ,inverse exits which is in G,for it to be abelian ,it means that it is ...
0
votes
2answers
55 views

Doubt pertaining to this Equivalence Relation.

$1$. True or false? If $R$ is an equivalence relation on a set $S$ and it has only finitely many equivalence classes altogether, then $S$ itself is a finite set. I think the answer is true ...
4
votes
2answers
108 views

Is $GL(2,\mathbb Z)$ a group?

Is the set of $2\times 2$ invertible matrices with integer entries a group under matrix multiplication? I believe not, because inverses for elements in this set may not be in the set (ie, may not have ...
1
vote
1answer
40 views

How many points does it take to identify a low-order polynomial in $\mathbb{Z}_N$?

I want to split the Bush's Baked Beans recipe into $M$ parts so that any set of $N<M$ people can reconstruct the recipe, but with the following constraints: Each person knows only a yes or no ...
1
vote
3answers
37 views

Find eight elements in $S_6$ that commute with $(12)(34)(56)$

This is a homework problem and I'm having trouble just getting a basic understanding. I understand that $S_6$ is a symmetric group of degree $6$. I'm not sure how to start looking for elements.
-2
votes
0answers
41 views

$R_P$ is a valuation noetherian ring

I need a hint to prove this question: Let $R$ be an integral domain and $P\neq 0$ a principal ideal of $R$ such that $\cap_{i=1}^{\infty}P^n=0$, show $R_P$ is a valuation Noetherian ring. I ...
1
vote
3answers
39 views

Inner product space over generalized number systems

Apologies for the lengthy setup, but I want to make sure I am clear on how I am using the notation, and what I mean by the phrase "generalized number system". Define a generalized number system $G$ ...
1
vote
1answer
55 views

K[$\alpha$]/K is algebraic

I have to prove that: If $\alpha \in E$ and $E/K$ is a field extension and $\alpha$ is algebraic over $K$ then $K[\alpha]/K$ is algebraic, but I do not know how to use the hypothesis to do this. Can ...
0
votes
0answers
23 views

On the Characterization of $F[\alpha]$ for a Field Extension $E/F$.

Let $E/F$ be a field extension. Reading a proof that $$ \alpha \text{ is algebraic over } F \implies [F[\alpha]:F] < \infty, $$ one might begin by considering an element $f(\alpha) \in F[\alpha]$ ...
0
votes
2answers
71 views

Factor $3x^{3/2}-9x^{1/2}+6x^{-1/2}$

I have $3x^{-1/2} (x^2-3x+2)$ However, I just tried to expand, and the answer is not the same as the original question. With fractional exponents I take out the smallest exponent, then I add the ...
0
votes
2answers
40 views

How to determine if a set is closed under some operation?

Is the set $\{-2,0,2\}$ closed under addition? And why? Specifically, when determining if a set is closed under an operation do you apply the operation to the each number and itself? For ...
0
votes
1answer
120 views

The group of bijections of an infinite set that move only finitely many elements

If someone can prove the following and give me an easy to understand explanation I would appreciate it! I have tried what omnomnomnom has suggsetd in the hints, but can't figure this out. A(S) is ...
2
votes
1answer
52 views

Is $F\subset F(a,b)$ a Galois extension?

I know that $F\subset F(a)$ and $F\subset F(b)$ are the Galois extensions of degree $n$ and $m$ respectively, $(n,m)=1$. 1) How can I show that $F\subset F(a,b)$ is the Galois extension either? 2) ...
6
votes
1answer
152 views

What is the correct statement of the infinitary associativity law?

Let $X$ denote a non-empty set. Write $\mathcal{L}$ for the class of all ordered pairs $(L,f)$ where: $L$ is a linear poset (possibly empty), and $f$ is an arbitrary function $L \rightarrow X.$ ...
0
votes
2answers
54 views

Greatest Common Denominator and linear combination

I know the gcd of 616 and 427 is 7, but I know need to do a linear combination of it. So there exists $x, y$ such that $$7=616x+427y$$ How do I solve for x and y?
1
vote
1answer
28 views

How to prove that: If two binary operations are anti-isomorphic and one of them is associative then the second one also will be associative?

We know what is called an anti-isomorphic operation on a set S. it is just a one two one $ g $ function mapping from $S$ to $S$. $ g: S \rightarrow S$. and it satisfy this condition $ g(xy)= ...
4
votes
2answers
63 views

Field Extension, Splitting Field and Galois Theory

Let $f(x)\in \mathbb{Q}[x]$ be an irreducible polynomial of degree $n\geq 3$. Let $L$ be the splitting field of $f$, and let $\alpha \in L$ be a root of $f$. Given that $[L:\mathbb{Q}]=n!$, prove that ...
9
votes
3answers
503 views

Does there exist a non-abelian group with roots?

I just learned the definition of a division group: An abelian group $G$ is called divisible if for every $x\in G$ and every $k\in\mathbb{Z}^+$, there exists $y\in G$ such that $y^k=x$ (or $ky=x$, ...
1
vote
2answers
41 views

If $\space o(\ker\phi) = n$, then $\varphi$ is an $n$-to-$1$ mapping from $G$ onto $\varphi(G)$

If $G,H$ are groups and $\varphi\colon G\to H$ a group homomorphism, how can I prove that if the order of $\ker\phi$ is $n$ then $\varphi$ is an $n$-to-$1$ mapping from G onto $\varphi(G)$
0
votes
2answers
140 views

Give an example of a field where -1=1

The question is to find a counterexample to the following: In every field $F$, $-1$ is not equal to $1$. My intuition leads me to integers modulo $1$. Is this correct, are the integers modulo $1$ a ...
0
votes
1answer
45 views

How do I start this proof?

How can I begin proving this? I am not sure where to start. A(S) is defined to be the set of all bijections from S to S.
1
vote
2answers
88 views

What is A Set Raised to the 0 Power? (In Relation to the Definition of a Nullary Operation)

In general an operation is a function $f:S^n \to S$, where $n$ denotes the arity of the operation, and a nullary operation is a function $f:S^0 \to S$. It is clear that $S^n=S \times S \times \cdots ...