# 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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### Pronunciation of Rng - the non-unital Ring

I chuckled the first time I heard that a Ring without a multiplicative identity (Ring without the i) is called a ...
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### Minimal polynomial in $T$-invariant subspace

I am stuck on the following problem. Problem: Let $V$ be a finite dimensional vector space over field $F$ and $T$ a linear transformation from $V$ to $V$. $W$ is an invariant subspace. Let $h_1$ be ...
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### A simple question about free group

Fix $r\in \mathbb{N}$ and let $\mathbb{F}_{r}=\langle g_{1}, ...,g_{r}\rangle$ be the rank-r free group. I have asked a question several days ago: Is $\mathbb{F}_{2}$ a subgroup of $\mathbb{F}_{3}$? ...
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### Suppose $H:= \{\sigma \in G| \sigma(1) = 1\}$, if for any $j \in \{1,2,…,n\}$ $t_j\in G$ such that $t_j(1) = j$. Show that $|G| = n|H|.$

Let G be a subgroup of the symmetric group $S_n$ in n letters. Consider the following subset of G: $$H:= \{\sigma \in G| \sigma(1) = 1\}$$ Suppose that G acts on the set $\{1,2,...,n\}$ transitively ...
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### elementary algebra question On the generator of a group

(Def): Let $G$ be a group and $X \subset G$. Let $\{ H_{\alpha} \}_{\alpha \in \Gamma}$ be a collection of all subgroups of $G$ which contain $X$. Then $\bigcap_{\alpha \in \Gamma} H_{\alpha}$ is ...
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### Prove $\text{rad}(I)/I$ is isomorphic to $\mathfrak{N}(R/I)$

I want to know if this is the correct way to do it. Define $\varphi:\text{rad}(I) \longrightarrow \mathfrak{N}(R/I)$ by $\varphi(r)= r^n+I$,then ker$\varphi = I$, so therefore by the 1st isomorphism ...
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### Multiplication in symmetric product space

STATEMENT: Let $V=\mathbb{R}^2$.Take $Y:=\left\{x\cdot y: x,y\in S_2(V)\right\}$ where $S_2(V)$ is the symmetric product of $V$. QUESTION: What is multiplication in the symmetric product space?
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### $F/E$ a finite Galois extension, the integral closure of $E[X]$ in $F(X)$

If $F/E$ is a finite Galois extension, then one can show that $F(X)/E(X)$ is also a finite Galois extension of the same degree (a basis for $F/E$ is also a basis for $F(X)/E(X)$). Since $E[X]$ is a ...
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### Question about the conjugation of an element in a group

Let $(a_1,a_2,a_3)$ be a 3-cycle in the alternating group $A_4$ in four letters. Find $g \in A_4$ such that $$g(a_1,a_2,a_3)g^{-1}=(a_2,a_1,a_4) = (a_1,a_2,a_4)^{-1}$$ Why do we need the last ...
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### Splitting field of $1 + x^3 + x^6 + x^9$

I'm studying for a qualifying exam and wanted to know if my reasoning on this problem was correct: Let $L$ be the splitting field of $f(x) = 1 + x^3 + x^6 + x^9$ over $\mathbb{Q}$. Find the Galois ...
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### Is $T:= \{g \in A_4|g^2 =(1)\}$ a subgroup of $A_4$?

Consider the subset $$T:= \{g \in A_4|g^2 =(1)\}$$ of the alternating group $A_4$ in four letters. Is T a subgroup of $A_4$? My Proof: Yes. If I am not wrong T is the Klein 4-group since only ...
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### Suppose that half of the elements of G have order 2 and the other half forma a subgroup H of oder n. Prove that H is an abelian subgroup of G.

Let $n>1$ be a positive integer. Let $G$ be a group of order $2n$. Suppose that half of the elements of G have order 2 and the other half forma a subgroup H of oder n. Prove that H is an abelian ...
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### What are some of the effective methods and theorems to prove that a group or subgroup is abelian? [on hold]

What are some of the effective methods and theorems to prove that a group or subgroup is abelian? Can someone give me a list of them based on your experience. Thanks. Now I have always been trying to ...
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### Trick with numbers, sums, cubes, squares? [on hold]

Let $(X_1, X_2, \ldots , X_n) = 10$ be a sum of positive numbers where $(X_1^2, X_2^2, \ldots , X_n^2) \geq 20$ is a sum of their squares. Prove that $(X_1^3, X_2^3, \ldots , X_n^3) \geq 40$.
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### Trying to calculate the quotient group $\mathbb{Z}\times\mathbb{Z}/\langle (1,1),(1,-1)\rangle$

Let $G$ be the group $\mathbb{Z} \times \mathbb{Z}$ and $H$ be the subgroup of $G$ generated by $(1,1)$ and $(-1,1)$. I am trying to calculate the quotient group $G/H$.
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### The number of solutions of $x^n = e$ in a finite group is a multiple of n, whenever n divides the group order.

Prove that in a finite group G the number of solutions of the equation $x^n = e$ is a multiple of n, whenever n divides the order of the group. I feel there is a very simple answer to this question, ...
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### Center of a ring projective?

If $R$ is a ring and $Z(R)$ denotes $R$'s centre, then when is $R$ projective as an $Z(R)$-module?
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### Universal property of the algebraic closure of a field

At page 4 of Strom's "Modern Classical Homotopy Theory" there is a universal formulation of the algebraic closure of a field. You can read it here from google books. Exercise 1.2a is then to convince ...
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### Set notation query

What do square brackets mean next to sets? Like $\mathbb{Z}[\sqrt{-5}]$, for instance. I'm starting to assume it depends on context because google is of no use.
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### Global dimension of the center

Let $R$ be a ring. Must the global dimension of the centre $Z(R)$ of the ring $R$ always be atmost that of $R$ itself? I mean is it generally true that: $D(Z(R)) \leq D(R)$ (where D is the global ...
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### can anyone give a proof by definition: $11$ is prime in $\mathbb{Z}[\sqrt{-5}]$

what i did is: assume $\alpha \notin (11),\beta\notin (11), \alpha\beta \in (11)\Rightarrow\exists \gamma, s.t.$ $\alpha\beta = 11 \gamma$, $\Rightarrow N(\alpha)N(\beta) = 11^2N(\gamma)$ then ...
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### Irreducibility over Q doesn't imply irreducibility over R

I want a counterexample of polynomial that is irreducible over $\mathbb Q$ but not irreducible over $\mathbb R$ (i.e not maximal over $\mathbb R$).
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### $|G| + \frac{|G|}{\left|\langle a\rangle\right|} + \frac{|G|}{\left|\langle b\rangle\right|} + \frac{|G|}{\left|\langle ab\rangle\right|}$

Show that for every finite group $G$ and for every elements $a, b \in G$ the following expression  |G| + \frac{|G|}{\left|\langle a\rangle\right|} + \frac{|G|}{\left|\langle b\rangle\right|} + ...