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.

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How to prove that a subgroup of a group is normal based on generating sets?

I apologize if this is a duplicate question, but I read online that one method by which to show that a subgroup is normal is by means of generating sets (if both groups have known presentations). In ...
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2answers
30 views

Isomorphism and Cardinality

Is the group $(G, *)$, where $G =\{(x, y) \mid x,y \in \mathbb{R}\}$ isomorphic to $(\mathbb{C}, *)$, where $\mathbb{C}$ is the complex numbers. My initial intuition says no since their are elements ...
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2answers
29 views

Order of a permutation

What does the order of a permutation actually mean? I accept the fact that it is the l.c.m. of the lengths of the cycles in its cycle decomposition, but I don't really have an intuition for what the ...
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1answer
37 views

Write Galois group as semidirect-product

Consider the polynomial $ x^7-13 \in \Bbb{Q}(x)$. Find Galois group and write it as a semi-direct product. Edit: So here is what I have done. I found the dimension of the splitting field over ...
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46 views

Realize groups as unit group of a ring

Let $A$ be a ring, $G$ be a group, and $f:A^{\times} \rightarrow G$ be a group homomorphism. Is there any ring $B$ and ring homomorphism $\varphi:A \rightarrow B$ such that $G$ is subgroup of ...
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2answers
21 views

Constructing group homomorphisms.

Let $G$ and $H$ be groups, and suppose I want to construct a group homomorphism $\phi$ between them. From what I know, I just need to send each element $x \in G$ to an element $y \in H$ such that the ...
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0answers
24 views

The Grothendieck group construction

In Lang, the example is stated as follows: Let M be a commutative monoid, written additively. There exists a commutative group K(M) and a monoid homomorphism. $$\gamma: M\rightarrow K(M) $$ having ...
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1answer
18 views

What is the usual definition of a zero divisor?

Let $R$ be a ring. I found there are two distinct definitions: Wikipedia Definition $a\in R$ is a zero divisor iff there exists nonzero $b\in R$ such that $ab=0$ or $ba=0$. Another: ...
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1answer
29 views

Normal extension and group of automorphisms

Let $K \supset F$ a normal extension (finite). Show in details that, for every $\alpha \in K$, the polynomial $$f(x)=\prod_{\sigma \in Aut_F(K)}(x-\sigma(\alpha))\in F[x];$$ My attempt: Since $K$ ...
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0answers
41 views

Property of group G with $|G|=2n$ with $n$ elements of order $2$ (Sylow theorem application)

Suppose $G$ is a group such that $|G|=2n$, $G$ has $n$ elements of order $2$ and the rest of the elements form a subgroup $H$. Show that $H \lhd G$ and $n$ is odd. I am pretty lost with this ...
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39 views

How do I get the generators of a group formed by combining two groups with known generators?

Consider two groups, $G$ and $H$, with generating sets $S$ and $T$, respectively. (That is, $G=\langle S \rangle$ and $H=\langle T \rangle$.) Let us say that we can represent elements of both $G$ ...
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75 views

Question regarding adjoint functors

Suppose $B \rightarrow A$ is a morphism of rings. If $M$ is an $A$-module, one can create $M_B$ by considering it as a $B$-module. This gives a functor $\cdot_B: \mathrm{Mod}_A \rightarrow ...
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1answer
17 views

Non-singular bilinear forms over a vector space.

I have a field $k$ and a finite dimensional $k$-vector space $E$. Let $f$ be a symmetric $k$- bilinear form on $E$. I define $f$ to be non-degenerate if $f(x,y)=0$ $\forall y\in E$ implies $x=0$. I ...
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0answers
32 views

Self normalising p sylow

When are p-sylow subgroups self normalising? I know, for example, that if the group has order $ p^2q^2$ then the p-sylow subgroups are self-normalising if there are $q^2$ of them. I just don't know ...
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1answer
36 views

Nonhomogeneous Linear Systems and Vector Space Solutions

Are there any nonhomogeneous linear systems with a solution set that forms a vector space? I know that, in order to be a vector space, a set must consists of a set V together with operations + (called ...
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1answer
51 views

adjoint of forgetful functor related to localization

Let $A$ be a ring and $S$ a multiplicative subset of $A$ such that $1 \in S$. Let $G$ be the forgetful functor from $Mod_{S^{-1}A} \rightarrow Mod_A$. Taking an $S^{-1}A$-module N and consider it as ...
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2answers
25 views

Proof of $g^a \equiv 1 \pmod{m}$ and $g^b \equiv 1 \pmod{m} \quad \implies \quad g^{\gcd(a,b)} \equiv 1 \pmod{m}$

I am trying to understand the following: $$g^a \equiv 1 \pmod m \quad \text{and} \quad g^b \equiv 1 \pmod{m} \quad \implies \quad g^{\gcd(a,b)} \equiv 1 \pmod{m}.$$ I have tried a few approaches, ...
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2answers
40 views

If $|G|<\infty$ and $H\leq G$ is such that $[G: H]=2$ then $|x^G|=|x^H|$ or $|x^H|=\frac{1}{2}|x^G|$ for all $x\in H$?

Let $G$ be a finite group and $H$ a subgroup of $G$ with index $2$, that is, $[G: H]=2$. Recall that $$C_H(x)=H\cap C_G(x), $$ where $C_G(x)=\{g\in G: gx=xg\}$. How can I use the second isomorphism ...
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1answer
22 views

Compute the following permutations [closed]

|(1254)|. I think they are trying to ask of the order of the permutation but I'm not sure how to solve it.
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1answer
44 views

Determining if a set is a group

Let $S=\lbrace x+y\sqrt2 : x,y\in \mathbb R \rbrace$ \ $\lbrace0\rbrace$. Justify whether $S$, together with traditional multiplication, is a group. I've verified that the set is closed under the ...
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0answers
29 views

In U(n) how do we find the element such that $x^2=1$ [closed]

For any integer n>2, show that there at least two elements in U(n) such that $x^2=1$
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2answers
115 views

Show that every maximal ideal in $ \mathbb{Z}[x, y] $ contains a prime number [closed]

Let $\mathfrak{M} \subseteq \mathbb{Z}[x, y]$ be a maximal ideal. Show that $ \exists\ p \in \mathbb{Z}$, $p$ prime such $p \in \mathfrak{M}.$ Thanks for the answers. I'd be interested in a proof ...
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1answer
40 views

Proof that $ \langle x \rangle = \{rx + nx \mid r \in R, n \in \mathbb Z\}.$

I need help in constructing a proof for this: Let M be an R-Module and $x\in M$. Then the submodule generated by x is given by $$\langle x \rangle = \{rx + nx \mid r \in R, n \in \mathbb Z\}.$$
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1answer
61 views

Complement but not direct summand; Lam, Lectures on Modules and Rings, Example 6.17(5)

Let $S$ be a submodule of an $R$-module $M$. A submodule $C⊆M$ is said to be a complement to $S$ (in $M$) if $C$ is maximal with respect to the property that $C∩S=0$. (This does exist by Zorn Lemma.) ...
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27 views

find these linear functionals

I'm trying to solve this question: My attempt of solution: If $x\in E$, see $x$ in the first $m$ coordinates of $\mathbb R^n$ (can we do this?). I know how to find linear functionals such that ...
2
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1answer
44 views

Goursat's Lemma proof

There is a problem in Lang's book that I don't quite understand how to proceed. It is problem #5, pg 75. I have already shown that the subgroups N and N' can be identified as normal in G, G'. But I ...
2
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3answers
52 views

Zero divisors and units of $\mathbb Z[X]/ \langle X^3 \rangle$

Problem: Find the zero divisors and the units of the quotient ring $\mathbb Z[X]/\langle X^3 \rangle$. If $a \in \mathbb Z[X]/ \langle X^3 \rangle$ is a zero divisor, then there is $b \neq 0_I$ ...
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1answer
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Are irreducible polynomials and irreducibles (in an integral domain) different?

There's this theorem that you can factor polynomials (in $\mathbb{Z}$[x]) into polynomials of lower degrees r and s in $\mathbb{Q}$[x] iff you can factor that polynomial into polynomials of the same ...
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1answer
28 views

$\varphi(f)$ is invertible iff $f$ is non-degenerate?

Let $E$ be the vectorial space of the bilinear functions $\varphi: \mathbb R^n\times \mathbb R^n\to \mathbb R$. Then, there is a canonical isomorphism between $E$ and the set of the real matrices ...
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28 views

Cubic resolvent of quartic.

Where does the cubic resolvent of quartic come from? I would like to know its derivation since I am having a hard time memorising the formulas.
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1answer
49 views

$\mathbb R[X] /<X^2-1>$ and $\mathbb R[X,Y]/<XY>$ are not fields

I have to prove that 1)$\mathbb R[X] /<X^2-1>$, and 2) $\mathbb R[X,Y]/<XY>$ are not fields. So, I must exhibit an element $r$ from say $\mathbb R[X] /<X^2-1>$ that has no ...
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1answer
22 views

Injective endomorphism on a finite field is surjective?

Can you guys give me any hint on how to prove(or disprove): any injective endomorphism on a finite field is also surjective?
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35 views

Isomorphism of a set [duplicate]

We know that $\operatorname {Aut}(G) \over \operatorname {Inn}(G)$ $\cong \operatorname {Out}(G)$. Is it true that $\operatorname {Aut}(G) \cong \operatorname {Inn}(G) \rtimes \operatorname {Out}(G)? ...
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1answer
46 views

automorphism group of groups [closed]

Given a group $G$, I would like to calculate $\operatorname{Aut}(G)$. From definition of $\operatorname{Aut}()$ we know: $\operatorname{Aut}(G)\le \operatorname{Sym}(G) $ If the group is finitely ...
3
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1answer
62 views

Number theoretic proof that $n\mid\phi(a^n-1)$

While 'playing' with the multiplicative group of integers mod $n$, I noticed that $n\mid \phi(a^n-1)$. The proof is straightforward: $a \in \left ( \mathbb{Z}/(a^n-1)\mathbb{Z} \right ...
3
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1answer
48 views

Transitive action on two sets! [duplicate]

Suppose $G$ is a finite group and G acts transitively on sets $X$ and $Y$. Let $a$ and $b$ belongs to $X$ and $Y$ respectively and $G_{a}$ be stabilizer of $a$ in $X$ and $G_{b}$ be stabilizers of ...
2
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1answer
53 views

Why the paired orbit has the same size here?

enter link description here On this proof, he just showed that the the paired orbit of (a,b) has the same size, but this seem has nothing to do with the size of corresponding paired orbit of an orbit ...
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75 views

Set containing all rings!

Does there exist a set containing all rings ? Possible idea :I think such set is not possible.If S is a set containing all rings i think we can again define a structure on S to make it Ring and that ...
2
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1answer
37 views

Ring homomorphism $f: \mathbb C \to \mathbb C$ such that $f(\mathbb R) \subset \mathbb R$

Exercise Find all the ring homomorphisms $f: \mathbb C \to \mathbb C$ such that $f(\mathbb R) \subset \mathbb R$ My attempt at a solution In a previous problem I've already proved that the only ...
3
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1answer
14 views

Every finite dimensional representation of an algebra has an irreducible sub representation

Let $V$ be a nonzero finite dimensional representation, i.e we have a homomorphism $\rho\colon A\rightarrow \text{End}_k(V)$, of an algebra $A$. I have to show that there is an irreducible sub ...
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The Freyd-Mitchell Embedding Theorem and projective (injective) objects

Given a small abelian category $\mathcal{A}$, the Freyd-Mitchell Embedding Theorem gives me a fully faithful exact functor $F:\mathcal{A}\rightarrow R$-$\mathsf{Mod}$, for some unital ring $R$, so ...
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1answer
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Statements about ring homomorphisms and division rings

Problem Decide whether the following statements are false or not. 1) If $A$ is a commutative ring such that every ring homomorphism different from the null morphism $\phi:A \to A'$ is injective, ...
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1answer
72 views

Some functorial maps $G\times G\rightarrow G$

Let $G$ be a group. Le diagonal map $\delta:G\rightarrow G\times G$ obviously gives a functorial morphism from the identity functor of $\mathbf{Grp}$ to the functor $P$ sending $G\mapsto G\times G$ ...
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43 views

Group of order $pq$ with $p\not\mid (q-1)$

Let $p, q$ be prime numbers, with $p<q$. If $G$ is a group of order $pq$ and $p\not\mid (q-1)$, then $G\cong \mathbb{Z}/pq\mathbb{Z}$. The standard way to prove this fact is using Sylow theorems, ...
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1answer
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Find all abelian subgroups of 4-element permutation group

I have to: "Find all abelian subgroups of a $4$-element permutation group $\Sigma_4$" I don't know what is $\Sigma_4$. Don't know how to bite this topic. The exercise seems too general to me. Any ...
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1answer
16 views

How do I compute the normaliser of a group G, considered as a set, in the group of set bijections of G?

Suppose G is a group and T(G) is the group of set bijections of G. I identify the elements of G as maps corresponding to the left multiplication by the chosen element. Then the normaliser of G in T(G) ...
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Crossed homomorphism: $\varphi(x)=\varphi(y)\iff Kx=Ky$

Let $G$ be a finite group, $N\unlhd G$. A crossed homomorphism from $G$ to $N$ is defined as a $\varphi:G\to N$ s.t. $\varphi(xy)=\varphi(x)^y\varphi(y)$. It's not in general a group homomorphism. ...
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1answer
30 views

How do I take the contraction of an ideal which is not in the image of the given morphism?

If I have a morphism of rings $\phi: A \to B$ which is not surjective, how should I take the preimage of an ideal not contained in the image of $\phi$?
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Let $a$ and $b$ be two elements in a commutative ring $R$ and $(a, b) = R$, show that $(a^m, b^n) = R$ for any positive integers $m$ and $n$.

I stumbled across a question that I have no idea how to start. I know the questions asking to show that the multiples of $a$ and $b$ as an ordered pair make still make the whole ring. Any sort of ...
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2answers
83 views

Field that contains $(\mathbb{Z}/p^n\mathbb{Z})^*$

Let $p$ be an odd prime. There exists a field $F$ that contains an isomorphic copy of $(\mathbb{Z}/p^n\mathbb{Z})^*$ as a multiplicative subgroup? Clearly if $n=1$ we can take $F=\mathbb{F}_p$, but ...