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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Subgroup H group G.

Let $G$ be a group, $H$ a subgroup of $G$, and $N:=\cap_{x\in G} \ \ x^{-1}Hx$. Prove, that $N$ is normal subgroup in $G$. I did this: Let $g\in G$. Whether $g(xhx^{-1})g^{-1}\in N$? Take $f=gx$. ...
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0answers
17 views

Isomorphism of a quotient group [duplicate]

I have that $G=S_4$ and $N = \{1, (12)(34), (13)(24), (14)(23)\}$, and thus far I have shown that N is a normal subgroup of G. I'm trying to figure out what group $G/N$ will be isomorphic to, but I ...
1
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0answers
21 views

Order of a certain finitely generated group

Suppose I am looking at the group $W=\langle s_\alpha, t_\beta \rangle$ where $s_\alpha$ and $t_\beta$ are reflections in $\mathbb{R}^2$ coming from two vectors $\alpha$ and $\beta$ making an angle of ...
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2answers
22 views

Is the set of all $f$ such that $\lim_{x\to1^-}f(x) = 0$ an ideal in the ring of functions from $[0,1]\rightarrow \mathbb{R}$?

Is the set of all $f$ such that $\lim_{x\to1^-}f(x) = 0$ an ideal in the ring of functions from $[0,1]\rightarrow \mathbb{R}$? I'm sure about the closure under addition but not quite clear about if ...
2
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1answer
29 views

Integer (or whole) numbers in arbitrary fields.

Given an arbitrary field $K$, may I define an integer in $K$? I have found how to define an algebraic number in $K$ and how to define an integer algebraic number in $K$. For instance, let ...
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6answers
79 views

Prove or give a counterexample: If a group G has a subgroup of order n, then G has an element of order n

The question is given in the title. I am unable to come up with a counterexample and I'm thinking this could apply to cyclic groups but not necessarily to general groups. Does anyone have any ideas or ...
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1answer
39 views

For a, b ∈ R we define a ∗ b := ab − a − b + 2 ∈ R. Furthermore let G := R\{1}. [closed]

For a, b ∈ R we define a ∗ b := ab − a − b + 2 ∈ R. Furthermore let G := R \ {1}. (a) Show that a ∗ b ∈ G for all a, b ∈ G. (b) Show that G together with the binary operation G × G → G, (a, b) |→ a ...
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1answer
31 views

Number of permutations with given cyclic structure

If $\sigma$ is a permutation made up by the disjoint cycles $\tau_1, \dots, \tau_r$ (including those of length $1$), we call structure of $\sigma$ $$(l_1, \dots, l_r),$$ where $l_1, \dots, l_r$ are ...
4
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0answers
61 views

Generalisation of euclidean domains

Recently I wondered how dependent the definition of euclidean domains is of the co-domain of the norm-function. To be precise: Let's define a semi-euclidean domain as a domain $R$ together with a ...
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1answer
22 views

Verify $(H_1 \cap H_2)g=H_1g \cap H_2g$ where $H_1,H_2\leq G$ and $g\in G$.

Verify $(H_1 \cap H_2)g=H_1g \cap H_2g$ where $H_1,H_2\leq G$ and $g\in G$. Let $S_1,S_2\leq G,g\in G$ What i had done, $x\in (S_1 \cap S_2)g$. Then $x=sg,s\in S_1\cap S_2$. So clearly, $x\in ...
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1answer
50 views

How to use the Universal Coefficient Theorem to determine $H^i(M; \mathbb{Z}_p)$ from $H^i(M; \mathbb{Z})$? [closed]

Let $M$ be a path-connected finite $CW$-complex. Suppose $$ H^2(M;\mathbb{Z})=\mathbb{Z}_{2k}, \text{ } k\geq 3; $$ $$ H^3(M;\mathbb{Z})=\mathbb{Z}\times\mathbb{Z}_{2}; $$ $$ ...
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2answers
45 views

An irreducible polynomial in a subfield of $\mathbb{C}$ has no multiple roots in $\mathbb{C}$

Let $K\subset \mathbb{C}$ be a subfield and $f\in K[t]$ an irreducible polynomial. Show that $f$ has no multiple roots in $\mathbb{C}$. If I understand this question correctly, I must show that ...
1
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1answer
66 views

Number of solutions in a field of order $32$ [duplicate]

Let $F$ be a field of order $32$. Then find the number of non-zero solutions $(a,b)\in F\times F$ of the equation $x^2+xy+y^2=0$. As , $|F|=32$ , so $(F\setminus\{0\},.)$ forms a group of order $31$, ...
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0answers
41 views

Monoids and groups

everybody. I got this exercise from Jacobson. Let $M$ be a monoid generated by a set $S$ and suppose every element of $S$ is invertible. Show that $M$ is a group. Proof: every element of $M$ has ...
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1answer
51 views

Group of order 6 contains an element of of order 3

I need to show that if $G$ is non-abelian group of order $6$ then it contains an element of order $3$. I don't know how to proceed. Any kind of help/hint is appreciated.
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1answer
29 views

$f$ is divisible by a square of non-constant polynomial iff $f,f'$ are not relatively prime

Let $R$ be a commutative ring and $f=a_0+ \cdots +a_nt^n \in R[t]$. Define $f':=a_1+2a_2t+ \cdots + na_{n-1}t^{n-1}$. Show that $f$ is divisible by a square of non-constant polynomial if and only ...
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2answers
21 views

Union Over a Totally Ordered Set of Ideals is an Ideal

I am trying to understand the proof of a theorem which uses Zorn's lemma. I understand quite well all parts of the proof except for one point: Let $R$ be a ring and define $K\doteq \{I\subseteq R ...
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1answer
26 views

Characterization of simple representations

I am trying to solve exercise 10 from chapter 2 of Peter Webb's book on representation theory: Prove the following theorem of Burnside: let $G$ be a finite group and let $k$ be a algebraically ...
2
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1answer
47 views

Turning Preordered Sets into Preordered Monoids (Constructing Preordered Monoids from Preordered Sets)

Question: Referring to the Wikipedia article on Adjoint Functors in Section 2 (Motivation), they talk about "turning rngs into rings" (can be rephrased as "constructing rings from rngs"). I do not ...
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1answer
46 views

Determine all the subgroups of the dihedral group $D_{15}$

Is there an algorithm for finding all of the subgroups of $D_{15}$? Also, is there a formula for finding the size of that subgroup? Not sure where to start with finding all the subgroups of $D_{15}$ ...
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1answer
24 views

Finding an orthonormal basis for $\mathbb{C}^2$ with respect to the Hermitian form $\bar{x}^TAy$

Problem Statement: Let $A=\begin{bmatrix} 2 && 1 \\ 1 && 2 \end{bmatrix}$. Find an orthonormal basis for $\mathbb{C}^2$ with respect to the Hermitian form $\bar{x}^TAy$. I am ...
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1answer
47 views

Prove that the set $U = \{(123), (124), … , (12n)\}$ can be used to generate $A_n$.

A hint is provided with the proof prompt: $(abc) = (1ca)(1ab)$, $(1ab) = (1b2)(12a)(12b)$, and $(1b2) = (12b)^2$. My idea: $(1ab) = (12b)(12b)(12a)(12b)$. To solve for the other half of $(abc)$, I'm ...
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1answer
43 views

showing non-isomorphism of groups [duplicate]

How do I prove that there is no isomorphism between $\Bbb Z$ under addition and $\Bbb Q$ under addition? They both are infinite order. I thought they might be isomorphic. Help would be ...
0
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1answer
49 views

Groups isomorphic to $S_{4}/N$

Let $G = S_4$ be a group, $N = \{1, (1, 2)(3, 4), (1, 3)(2, 4), (1, 4)(2, 3)\}$ a normal subgroup of G. It's easy to see that $G/N$, the set of cosets is $G/N = \{a, b, c\}$, where $$a = \{(1), (1, ...
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0answers
41 views

Finding homomorphisms in normal groups to $S_n$

This question is related to my trouble in abstract algebra in general so some general advice is really appreciated!! (I'm really new to this.) Let A be an infinite subset of G with $n = \#A$. ...
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1answer
28 views

If $\sigma$ is a product of $k$ transpositions, prove that $k \equiv$ inv $(\sigma)$ (mod $2$).

An inversion in a permutation $\sigma = \sigma_1...\sigma_n$ is a pair of indices $i < j$ such that $\sigma_i > \sigma_j$. Let inv($\sigma$) be the number of inversions of $\sigma$. If ...
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2answers
45 views

Showing a non-isomorphism of groups

I need to show that $\Bbb Z^*_8$ is not isomorphic to $\Bbb Z^*_{10}$. $\Bbb Z^*_n$ means integers up to $n$ coprime with $n$ I do not know how to do this. I have difficulties doing proofs ...
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1answer
30 views

Does localization of a Noetherian ring always give a local ring? [closed]

I have a local ring $A$ and suppose I localized this ring at prime $P$. Is the localized ring $A_P$ a local ring? I was wondering if it requires additional properties on $A$. Thank you very much!
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34 views

Every non-Noetherian module has a submodule maximal with respect to being not finitely generated. [duplicate]

Let $M$ be a module. Show that if $M$ is not Noetherian then $M$ has a submodule $N$ such that $N$ is not finitely generated whenever $N<A\leq M$. The question is related to If $M$ isn't ...
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1answer
59 views

Property of a Noetherian ring: How come $P \setminus P^2$ is non-empty? ($P$ is a prime ideal) [closed]

Let $A$ be a Noetherian ring, and let $P$ be a prime ideal. How come we know that $P \setminus P^2$ is non-empty? Thank you!
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1answer
24 views

Structure of induced module for subgroup $H \le G$, confused by mixing up notions of $FH$-submodule and $F$-subspace

The following is a question on a passage in the book A Course in the Theory of Groups by Derek Robinson. There he constructs the induced module $M^G$ from a $FH$-module for some subgroup $H \le G$ of ...
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3answers
31 views

Two disjunct normal subgroups

Let M, N be normal subgroups of G with M∩N={e}. I'm trying to prove that MxN is isomorphic to G. I proved that nm=mn for all n in N and m in M. So now I'm trying to take any fixed g in G and ...
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1answer
30 views

Determining all the homomorphisms $\mathbb{Z} \to R$, where R is an integral domain.

I think I have this question figured out almost completely, but I'm a little worried about using a certain notation. Suppose $\mathbb{Z} \stackrel{\phi}{\longrightarrow} R$ is a ring homomorphism. ...
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1answer
41 views

field of fractions of $k[X]$ [closed]

Let $k$ be a field and suppose $$k(X)=\text{field of fractions of }\ k[X]=\left\{ \frac{f(X)}{g(X)}\mid f,g\in k[X], g\neq 0\right\}.$$ Show that $k(X)$ is not a finitely generated $k$-algebra.
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1answer
31 views

homomorphism of profinite groups

Let $G$ be a profinite group and consider a continuous surjective homomorphism: $$\phi:G\rightarrow \widehat{\mathbb Z}$$ where $\widehat{\mathbb Z}:=\varprojlim \mathbb Z/n\mathbb Z$. Moreover Let ...
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1answer
73 views

When is an ideal also a ring, and could then be anything said about its relation to the original ring

If $R$ is a ring with unity $1$, then $S \subseteq R$ is called a subring if it is itself a ring with $1 \in S$. A subset $I \subseteq R$ is called an ideal if it is a group with respect to addition ...
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0answers
23 views

Show that we have a ring isomorphism $\varphi : D^{op} \rightarrow {End_{M_n {(D)}}}(D^n) $. [closed]

I am trying to solve the following Representation Theory question: Suppose that $d \in D$ and define the map $$ \varphi_d \colon D^n \rightarrow D^n $$ by $$ \varphi_d((v_1, \ldots, v_n)) ...
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1answer
70 views

The set of isomorphisms from a right coset of the automorphism group $Aut(X)$ in $S_n$.

From "Lecture Notes in Computer Science" by Christoph M. Hoffmann , on page 22- Theorem 4 Let $X$ and $X'$ be two isomorphic graphs with vertex set $V = \{1 ..... n\}$ , Then the set of ...
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1answer
28 views

Structure theorem for non connected graded Hopf algebras

Let $V^{\bullet}$ be a graded vector space. Assume that $V^{i}=0$ for $i<1$. Then the tensor algebra has the structure of a connected Hopf algebra and in particular its Lie algebra of primitive ...
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1answer
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In proof of $|G| = n_1^2 + n_2^2 + \ldots + n_h^2$ how could equality of dimensions concluded from ring isomorphism

In Derek Robinson, A Course in the Theory of Groups on page 224 he proves: Let $G$ be a finite group and let $F$ be an algebraically closed field whose characteristic does not divide the order of ...
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2answers
51 views

motivation for the direct limit [closed]

I know just the very basics on Category Theory and that's why I'm going to ask a stupid question. I'm trying to get an intuition for direct limits for my course on Commutative Algebra. All the books ...
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2answers
28 views

Which one of the below options is correct?

I think the option $(Q)$ is true since $O(Q/\{-1,1\})= 8/2 = 4 = 2^2$. Since order is $p^2$ thus $(Q)$ option is true. Can anyone suggest about option $(P)$? Thanks
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0answers
25 views

Lang's proof of the Weierstrass preparation theorem

Relevant Google Books link. I'm having problems with the final step in the proof of Theorem 9.1. It's not clear to me why the function $I + \tau \circ \frac{\alpha(f)}{\tau(f)}$ should be ...
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0answers
45 views

Field of fractions of $\mathbb{C}[X,Y,Z]/\langle XY-Z^2\rangle$ [closed]

Let $R=\mathbb{C}[X,Y,Z]/\langle XY-Z^2\rangle$ and let $\mathbb{C}(X,Y)$ be the field of fractions of $\mathbb{C}[X,Y]$. Show that the field of fractions of $R$ can be expressed as ...
2
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1answer
48 views

Producing a $G$-invariant form from the standard Hermitian product using the averaging process

Problem statement: Let $G$ be a cyclic group of order $3$. The matrix $$A=\begin{bmatrix} -1 && -1 \\ 1 && 0 \end{bmatrix}$$ has order $3$, so it defines a matrix representation of ...
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43 views

Show $SO_2(\mathbb{R}) \cong\{A \in GL_2(\mathbb{R}) : A^tA = I, \det A = 1\}$.

Show $SO_2(\mathbb{R}) \cong\{ A \in GL_2(\mathbb{R}) : A^tA = I, \det A = 1\}$, where $SO_2(\mathbb{R})$ is the group of rotations of the circle under the operation of composition. Attempt: ...
1
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1answer
27 views

Does every non-empty quasigroup have a left or right identity?

I know that some quasigroups are not loops, meaning they don't have a two-sided identity. But are there non-empty quasigroups that don't even have one-sided identities?
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0answers
17 views

Prove that $\varepsilon(v) \equiv \varepsilon(u) \equiv 1 (2)$

Suppose I have a finite group $G$ and its integral group ring $\Bbb{Z}G$. Let $P < G$ , thus we have $\Bbb{Z}[C_G(P)] \subseteq \Bbb{Z}G$. Let $u\in U(\Bbb{Z}G)$ and let $v\in \Bbb{Z}[C_G(P)]$ be ...
1
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1answer
41 views

Prove that $g\in P$

Let $G$ be a finite group and $P$ a Sylow $2$-subgroup. If $g$ is an element of $G$ with order a power of $2$ then it lies in some conjugate of $P$. I get this. But if also $g\in C_G(P)$, then it is ...
0
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0answers
57 views

Proving this fact about algebraic sets.

I want to prove the following equivalence: Let $V$ an algebraic set, $K$ a field and $\overline K$ its algebraic closure. Then we say that $V/K$ ($V$ is defined over $K$) if $I_{V}$ (the ideal ...