# Tagged Questions

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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### Does the commutator group of $S_n$ equal $A_n$ in general?

And how would one deduce this? $[S_n, S_n]$ consists of even permutations so it's obvious that $[S_n, S_n] \leq A_n$, but is $[S_n, S_n] = A_n$ true as well? If so, how to deduce this? If not, how ...
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### Showing a finite abelian group is cyclic assuming something about all homomorphic images of it

Let $G$ be a finite abelian group such that $|G|\ne p^n$ for any prime $p$. If every homomorphic image $\varphi (G)$ with $|\varphi (G)| < |G|$ is cyclic, then show $G$ is cyclic. This is an old ...
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### Prove that $\Bbb{R}[\cos(\theta),\sin(\theta)]\cong\Bbb{R}[x,y]/(1-x^2-y^2)$ [duplicate]

More precisely, given the ring homomorphism $\phi:\Bbb{R}[x,y]\to\Bbb{R}^\Bbb{R}$, with $\phi(f(x,y)):\Bbb{R}\to\Bbb{R},\,\,\phi(f(x,y))(\theta)=f(\cos(\theta),\sin(\theta))$, where $\Bbb{R}[x,y]$ is ...
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### The group algebra $KG$

If $G$ is a cyclic group of order $m$. Then $KG\cong K[t]/(t^m-1)$. Where $K$ is a field. I define \begin{align*} \varphi:K[t]&\longrightarrow KG\\ \sum_ia_it^i&\longmapsto\sum_ia_ig^i \end{...
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### What are the units of $U(\mathfrak{sl}_2)$?

Let $U(\mathfrak{sl}_2)$ be the Universal Enveloping Algebra of $\mathfrak{sl_2}$ over a field $K$, i.e. the (non-commutative) algebra generated by three generators $E,F,H$ subject to the commutator ...
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### Prove that any finitely generated submodule of $R^+$ (the field of quotients) is free of rank $1$

I am working on the following problem: Let $R$ be a principal ideal domain and $R^+$ the field of quotients. Then $R^+$ is an $R$-module. Prove that any finitely generated submodule of $R^+$ is a ...
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### $\mathbb{Z}_p$ is an Integral Domain

Assume that we define the ring of the $p$-adic integers as the projective limit $$\mathbb{Z}_p =\varprojlim \frac{\mathbb{Z}}{p^n\mathbb{Z}}$$ Then $\mathbb{Q}_p$, the field of the $p$-adic numbers ...
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### Mandelbrot set and times tables

I recently saw a mathologer video on YouTube titled Times tables, Mandelbrot set and the heart of mathematics. It was about generating patterns using tables of numbers. I don't have any idea about it. ...
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### Given a homomorphism defined on a generating set of a group how to define it for a general element?

Let $\mathbb Z^n$ and $\mathbb Z^d$ be free $\mathbb Z$-modules with $d>n$. Suppose $v_1,\dots,v_d$ are primitive vectors in $\mathbb Z^n$. Let $e_1^*,\dots, e^*_n$ be the dual basis (that is, a ...
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### Number of subgroups equal to order of group

Here is a fun question. Consider the dihedral group $\mathcal{D}_4=\left\langle a,b\mid a^4=b^2=1, bab=a^{-1}\right\rangle$ of order $8$. This group has exactly $8$ genuine subgroups (but not all ...
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### Are there groups $G$ isomorphic to $\mathrm{Aut}\left( \mathrm{Aut} (G)\right)$, such that $G\not\cong \mathrm{Aut}(G)$?

Trivial examples for the first condition are easy to find: $G={1},C_2$. Are there any (finite\non-finite) groups that satisfy the conditions in the title?
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### Can the dihedral groups be seen as subgroups of each other?

I am solving one difficult problem and now I need information on this: If $m\mid n$, is then possible to "embed" $D_m$ to $D_n$, or otherwise said, does $D_n$ have a subgroup isomorphic to $D_m$? ...
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### Algebraic structure on any infinite set

Given any algebraic object $X$, say group, ring, integral domain, etc., and a special subset $I$ of $X$ namely normal subgroup, ideal etc., it is always possible to put a structure on $X/I$ induced ...
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### Quotient ring with reducible polynomial

Let $S = \mathbb{R}[x]/(x^2+1)^2).$ The first goal is to show that there exist exactly two homomorphisms $\pi\colon S\to \mathbb{C}$ such that $\pi|_{\mathbb{R}} = \text{id}_{\mathbb{R}}.$ I know ...
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### Factoring $p(x) = x^n -1$ for any natural number $n$

Can I say that from inspection, $(x-1)$ is a factor which implies that $p(1) = 0$. I then used long division which then gave this: $p(x) = (x-1)(\sum \limits _{i=1} ^{n} x^{n-i})$ How would you have ...
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### I am confused about this notation for matrix representation.

I am confused about this notation from the image below. How can you represent something like (1,2),1 as a 2D matrix? For S1 = (0,1,2) and S2 = (0,1) I would expect two matrices like: [(1,0),(1,0),...
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### $17\mathbb{Z}[\sqrt {10}]$ is prime ideal in $\mathbb{Z}[\sqrt {10}]$

This seems tedious since I would have to show $(a+b\sqrt {10})(c+d\sqrt {10})=17k+17j\sqrt {10}$ implies that one of the factors belongs to $17\mathbb{Z}[\sqrt {10}]$. It's easier to show something ...
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### Show that $V=\mathbb KG\oplus\cdots\oplus \mathbb KG$.

Let $V$ a $\mathbb KG$-module such that the character of $V$ is such that $\chi_V(g)=0$ for all $g\in G\setminus \{1\}$. Show that there is an $m$ s.t. V=\underbrace{\mathbb KG\oplus\cdots\oplus \...
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### Properties of a subgroup of a group $\mathbb Z_p \times \mathbb Z_p$
Let $p \geq 5$ be a prime. Thhen which one of the followings are true. 1) $\mathbb Z_p \times \mathbb Z_p$ has atleast five subgroup of order p. 2) Every subgroup of $\mathbb Z_p \times \mathbb Z_p$ ...