Tagged Questions

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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What would be a counterexample of $N_G(T)\not\subset N_G(S)$?

Let $G$ be a group and $S,T$ be subgroups of $G$ such that $S\subset T$. Is there an example such that $N_G(T)\not\subset N_G(S)$? Also, what is an example such that $N_G(S)\not\subset N_G(T)$?
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1answer
8 views

a problem about normal extensions & automorphisms

this is my problem: Suppose $K|F$ is a normal extension,prove that for every $\alpha ,\beta \in K$ that have the same minimal polynomial over $F$,there is a F-algebra automorphism over ...
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0answers
5 views

Does every algebra automorphism preserve augmentation ideal filtration?

Let $A=\displaystyle\bigoplus_{n\geq0}A_n$ be a graded algebra and let the augmentation homomorphism $\varepsilon:A\to A_0$ be the projection. Define the augmentation ideal, denoted by $A_+$, to be ...
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Normal basis theorem

Let $K$ be a finite Galois extension of, say, $\mathbb{Q}$. Then is known(and called normal basis theorem) that if i view $K$ as a representation of $Gal(K/\mathbb{Q})$ over $\mathbb{Q}$ it is ...
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1answer
26 views

Group of order $18$ contains exactly one subgroup of order $9$

I'm trying to prove the following: Proposition: A finite group $G$ of order $18$ has a unique subgroup of order $9$. Here is my attempt: Observe that $18 = 3^2 \times 2$. Let's count the number ...
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1answer
25 views

Simple question on tensoring by a quotient ring

$A \subset B$ is an extension of commutative rings s.t. $B$ is a f.g. free $A$-module of rank $n$, so I have $A^n \stackrel{\sim}{\longrightarrow} B$ as $A$-modules. Let $\mathfrak a$ be an ideal of ...
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0answers
7 views

Finding the 2D irreducible representation of quaternions $Q_8$ in the space of functions $f\colon Q_8 \rightarrow \mathbb{C}$

The space of functions $F=\{f\colon Q_8\rightarrow\mathbb{C} \}$ is 8 dimensional, since we can choose for each element of $Q_8$ an element in $\mathbb{C}$ to send it to. The action of $Q_8$ on this ...
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17 views

Find a bijection between cosets

Let $G \supseteq H \supseteq K$, G Group with subgroups H and K. I want to show that $$\phi : R_{G/H} \times R_{H/K} \rightarrow G/K, (x,y) \mapsto x y K$$ is a bijection, where $R_X$ is set that ...
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0answers
10 views

showing that $G$ is super solvable.

every finite group that its order is square free ,have a non-trivial characteristic sylow subgroup and then it is super solvable. my problem is just in showing that it is super solvable, I want to ...
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1answer
31 views

Equivalence classes of extensions of ${\bf Z}_m$ by ${\bf Z}$

Problem : What is the set of equivalence classes of extensions of ${\bf Z}_m$ by ${\bf Z}$ ? Try : Note that $$ {\rm Ext}_{\bf Z}^1 (A:={\bf Z}_m,N)=N/mN $$ where $N$ is an abelian. From definition ...
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1answer
16 views

Simple Maximal Ideal Question.

Question: Let $R = \{a + bi | a,b \in \Bbb Z \}$, let $M = \{x(2 + i) | x \in R\}$. Prove M is a maximal ideal of R. I just started learning about ideals so I apologize for asking a basic question, ...
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2answers
18 views

a problem about splitting field & irreducibility of a polynomial

suppose that $K$ is the splitting field of $f(x)\in F[x]$ ,when the degree of $f(x)$ is $n$ & $[K:F]=n!$.show that $f(x)$ is irreducible over $F$. i know that $K|F$ is normal,but i don't know how ...
3
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1answer
29 views

Simple group with Klein four

If $G$ is a simple group, with a Sylow $2$-subgroup isomorphic to the Klein four group $\mathbb{Z}_2 \times \mathbb{Z}_2$, then I want to show that any two involutions in a given Sylow $2$-subgroup ...
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2answers
48 views

What is this group explicitly?

Let $G$ be finite group act on a set $X$ transitively. I already proved the set $\{ f : X \to X | f(g*x) = g*f(x) \, \forall x \in X, g \in G \}$ is a group. My question is what is this group ...
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2answers
23 views

Proving the existence of a homomorphism $\overline f:G/H\rightarrow G'$ such that $\overline f \pi = f.$

I'm working on a problem where I'm given that $G$ is a group, $H$ is a normal subgroup of $G,$ $f:G\rightarrow G'$ is a homomorphism, and $H\subseteq \ker(f).$ I need to show that there exists a ...
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1answer
86 views

If a finite group acts transitively on a set, does its center also acts transitively?

If $G$ is a finite group acts transitively on a set $X$. Does the center $Z(G)$ also acts on $X$ transitively? I don't see how this statement will be true but I can't come up with a counter example ...
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52 views

Does a group with $|G| = 33$ have to contain an element of order $11$?

A group with $|G| = 33$ must contain an element of order $11$. Prove or disprove. This is inspired by another MSE question. So we know that there must be an element with order 3. I tried using ...
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0answers
28 views

In $\mathbb Z[x]$, show that the only common divisors of $2$ and $x$ are $1$ and $-1$.

In $\mathbb Z[x]$, show that the only common divisors of $2$ and $x$ are $1$ and $-1$. $\mathbb Z[x]$ is the ring of poloynomials with integer coefficients. This should be a pretty trivial question. ...
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1answer
17 views

Orthogonal transformation between vectors of the same norm

Suppose $V$ is a vector space over a field not of characteristic $2$, and is equipped with an inner product. I want to show that, given vectors $v$ and $w$, there is some orthogonal ...
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2answers
28 views

Do the isomorphism's of groups form an equivalence relation on the class of all groups?

An isomorphism is simply a bijective homomorphism. How would one show that isomorphism's are symmetric, reflexive, and transitive?
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1answer
23 views

Show that if $G$ is cyclic then so is $H$

If group $G$ is isomorphic to group $H$, show that if $G$ is cyclic then so is $H$. An isomorphism is simply a bijective homomorphism. The latter is a function which preserves the group operation. ...
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1answer
11 views

Proof of ideals and the Chinese remainder theorem

Let $I$ and $J$ be ideals of a ring $R$. Prove that the pair of congruences $y\equiv r\,\mathrm{mod}\,I$ and $y\equiv s\,\mathrm{mod}\,J$ has a solution if and only if $r\equiv s\,\mathrm{mod}\,\, ...
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1answer
22 views

Group Homomorphism and injective

Given a group homomorphism $\psi:A_8\rightarrow S_9$ for which there exists $\alpha\in A_8$ with $\psi(\alpha)=(1\, 2)$, prove that $\psi$ is injective.
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1answer
14 views

Finding Sylow p-subgroups

I am trying some examples of finding Sylow p-subgroups in specific groups and looking for the most efficient way to do so. For example, lets say we need to find Sylow-3 subgroups in $A_4$ and $D_6$, ...
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1answer
24 views

Is it true that $n_p!\le |G|$?

Let $G$ be a finite group and $n_p:=|\text{Syl}_p(G)|$. Is it true that $n_p!\le |G|$ ? I've shown that it's true, but I'm not so sure, can you check my proof? Proof. Let $G$ act on ...
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2answers
32 views

Proving that the isomorphism is homomorphic.

I am asked to find an isomorphism from the group $G = 1,i,-1,-i$ to the group $H = \begin{bmatrix} 1 & 0 \\[0.3em] 0 & 1 \end{bmatrix} , \begin{bmatrix} ...
2
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1answer
18 views

Prove that group $G$ is abelian when $K$ field has only 2 elements

Let $K$ be a field and $G$ is a group. $G=\{(g,a) : g\in K, a \in K^*\mid (g,a)(h,b)=(g+ah,ab)\}$ $K^*$ means $K$ without ${0}$. Proove that $G$ is Abelian $\Leftrightarrow$ $K$ has only 2 elements. ...
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23 views

Group Action and Orbits

I am looking at the following example which says find the orbit of $0$ under addition by $2$ and $3$ if $\mathbb{Z}_4$ acts on itself by addition. So to find the orbit of $0$ we are looking at the set ...
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Is there a group which has precisely all finite groups as subgroups?

I would like to ask the following question: Does there exist a group $G$ such that every finite group can be embedded in $G$, and every proper subgroup of $G$ is finite? The closest ...
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0answers
37 views

$G/H$ contains element of order $n$ but $G$ does not

I'm trying to come up with a group $G$ and normal subgroup $H$ of $G$ such that $G/H$ contains an element of order $n$ (for some integer $n$), but $G$ does not. Does $G = \mathbb{Z}$ and $H = ...
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1answer
30 views

Kernel and image of homorphism

Q: Let $m$, $n$ be natural numbers. Suppose $m\mid n$. Define $\theta\colon \mathbb{Z}_n \to \mathbb{Z}_m$ by $\theta([a]_n)=[a]_m$. What is the kernel and image of $\theta$ I know what the general ...
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2answers
30 views

Solution verification: $G$ and $G/H$ contain elements of same order

I just took my abstract algebra midterm, and was wondering if someone could confirm my solution to the following problem. Problem: Let $G$ be a finite group and let $H$ be a normal subgroup of ...
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1answer
19 views

Is there a way to know which elements are a generator in a group?

$U_7 = {[1],[2],[3],[4],[5],[6]}$ This group with respect to multiplication. I know that $[3]$ is a generator; I verified this using trial and error. I wonder if there is a more systematic way to ...
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2answers
26 views

In $\mathbb{Z}_q$ where $q$ is prime, show that $[a^q]=[a]$ for all $[a]\in \mathbb{Z}_q$

Question: In $\mathbb{Z}_q$ where $q$ is prime, show that $a^q=a$ for all $a\in \mathbb{Z}_q$. My attempt: To show $[a^q]=[a]$ for all $[a]\in \mathbb{Z}_q$, it suffices to show that for any ...
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2answers
30 views

Show that $\deg(fg) = m+n$

Let $R$, a ring with a $1$ and $f,g$ two polynomials, where $\deg(f)=n, \deg(g)=m$. Also, there's a $c\in R$ such that $b_mc = 1$. Show that $\deg(fg)=m+n$. I'd be glad for a guidance. Thanks
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1answer
17 views

How to prove 120 degree rotations of a hexagon form a subgroup

Let H={$\rho_{0}, \rho_{2}, \rho_{4}$}, a subgroup of D6, the group of symmetries. Where $\rho_{0}$=identity permutation, $\rho_{2}$=(1,3,5)(2,4,6) and $\rho_{4}$=(1,5,3)(2,6,4) Identity is easy to ...
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0answers
26 views

Find the generating set $\langle S\rangle$.

Let $S = \{4,22\}$, a subset of $U(\Bbb Z_{35})$. Find $\langle S\rangle$. I have no idea where to start or what to do.
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1answer
22 views

Determine all the generators of $\mathbb{Z}_{25}^{\times}$

Determine all the generators of $\mathbb{Z}_{25}^{\times}$. Is there some way that I can use the fact that $\mathbb{Z}_{25}^{\times}$ is cyclic generated by $3$?
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1answer
31 views

Question on abstract algebra about Group?

I need an explanation, why $ (\mathbb{Z}_7,\oplus _6 )$ is not a Group? As I have discovered so far. The following conditions are satisfied I) Closed! II) Associative! III) ...
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1answer
28 views

Example of such groups [duplicate]

Does there exist $G$ such that for a subgroup $H$ of $G$ , $gHg^{-1}$ is proper in $H$ for some $g\in G$ ? It is clear that $H,G$ must be infinite. I look for examples in matrice groups and not ...
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0answers
16 views

Every finite group of square-free order is soluble

prove that every finite group which it's order is square free is soluble. I think it is enough to show that every sylow subgroup of this is cyclic. please tell me if my idea is right and if it is ...
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1answer
27 views

Abstract Algebra Groups of Order 2p

Groups of order 2p, where p is an odd prime. Suppose that G an element of order 2p. Prove that G isomorphic to Z2p. Hence G is cyclic. I can not use Sylow's theorem though since it has not yet been ...
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1answer
18 views

making an injective resolution for $A$.

suppose $A^{'}$ is sub module of $A$ and $$0\rightarrow A^{'}\overset{(d^{-1})^{'}}{\rightarrow}(I^{0})^{'} \overset{(d^{0})^{'}}{\rightarrow} (I^{1})^{'}\rightarrow \ldots $$ is injective resolution ...
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2answers
26 views

Determine whether $\phi$ is an automorphism

Let $H$ be a proper subgroup of $G$ and let $\psi$ be an automorphism of $H$ other than the identity mapping. Define a mapping $\phi:G\rightarrow G$ by $\phi(x)=$\begin{cases}\psi(x) \text{ if } ...
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0answers
41 views

$2$-groups with odd permutations

If $P$ is the Sylow $2$-subgroup of a finite group $G$, $H <P$, and $x \in P$ so that no non-trivial element of $\langle x \rangle$ conjugates into $H$ (in $G$), and $|P|=|H||x|$, how can I show ...
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2answers
26 views

Homomorphism well defined

Q: Let $m,n$ be natural numbers. Suppose $m\mid n$. Define $\theta\colon \mathbb{Z}_n \to \mathbb{Z}_m$ by $\theta([a]_n)=[a]_m$. Show that theta is well defined. I know if $m\mid n$ then there ...
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1answer
32 views

Determine whether $\phi$ is a homomorphism

Let $\phi: \mathbb{Z}_6 \rightarrow \mathbb{Z}_2$ be given by $\phi(x)=$the remainder of $x$ when divided by $2$, as in the division algorithm. Let $\phi: \mathbb{Z}_9 \rightarrow ...
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1answer
9 views

If $H\trianglelefteq G$ and$P$ is a $p$-Sylow subgroup of $G$ then $\gcd([H: H\cap P], p)=1$?

Let $G$ be a group, $H\trianglelefteq G$ and $P$ a $p$-Sylow subgroup of $G$. How can I show $$\gcd([H: H\cap P], p)=1?$$ Thanks.
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1answer
22 views

Let $I$ be a proper ideal of a ring $R$. Then $IR[\alpha_1, … , \alpha_n]$ is a proper ideal of $R[\alpha_1, … , \alpha_n]$

Let $I$ be a proper ideal of the commutative ring $R$. Then $IR[\alpha_1, ... , \alpha_n]$ is a proper ideal of $R[\alpha_1, ... , \alpha_n]$ I thought of using the fact that an ideal of any ring ...
1
vote
1answer
45 views

What is a good way to compactly write that a number is an integer between a and b?

Specifically, I refer to the following set: $$ \left\{ x\in\mathbb{Z}\mid a\leq x\leq b\right\} $$ where $a\in\mathbb{Z}$ and $b\in\mathbb{Z}$ such that $a<b$. Alternatively, this can be written ...