# Questions tagged [abstract-algebra]

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57,551 questions
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### Divisibility in Integers and General Rings [duplicate]

If x|y and y|x in integers, does this imply x=y? Similarly, for any general ring, if x|y and y|x in the ring, does this imply x=y?
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### Is $D_n$ isomorphic to $C_n \times C_2$?

I am doing a proof for an exercise where if the dihedral group $D_{n}$ $\cong$ $C_{n} \cdot C_{2}$ where $\cong$ means isomorphic and $\cdot$ is the direct product, I would be able to finish it, but I ...
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### Growth rate of free nilpotent group of rank $r$ and nilpotency index $s$.

Let $F^{(r)}$ denote the free group of rank $r$ with generators $x_1, \dots, x_r$. Recall a group $G$ is nilpotent of index $s$ if $G_{s+1} = \{e\}$ and $G_s \neq \{e\}$ (where $G_i$ denotes the ...
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### Factorization of a polynomial in $\Bbb F_7$

I need to reduce as much as possible the polynomial: $x^6+3x^5+2x^4+6x^3+4x^2+5x+2$ in the finite field $\Bbb F_7$. It has no roots over the field, and I don't think that it is necessary to check ...
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### Group is isomorphic to opposite group. [on hold]

Given a group G, define a new group G^op having the same set of elements, but a new binary operation ⊙ given by a ⊙ b = ba (where, on the right, we are computing the product of b and a, in that order, ...
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### Showing that $[G:K]=[G:H][H:K]$ for $K \leq H \leq G$ and $|G| < \infty$

Definition: For a finite group $G$ and a subgroup $H \leq G$, $$[G:H] := \frac{|G|}{|H|},$$ which is a positive integer. For a finite group $G$, assume $K$ is a subgroup of $H$ and $H$ is a ...
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### Why is this map surjective?

The author proves that there are two isomorphism classes of groups of order 21: the class of the cyclic group $C_{21}$, and the class of a group $G$ generated by two elements $x$ and $y$ that satisfy ...
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### Bounding the exponent in unit groups of modular group algebras

Let $p$ be a prime number, $G=:G_0$ a (non-Abelian) finite p-Group and $K$ a finite field with $\operatorname{char}(K)=p$. It is well-known that the group $G_1:=1+\operatorname{rad}(KG)$ is a p-group ...
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### Define dimension without referring to bases

The dimension of a vector space is the common cardinality of all bases. I would like to define it in a way that does not refer to bases. I only care about finite dimensional vector spaces. First, I ...
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### Product of characters that is not irreducible

I am looking for an example of product of irreducible characters that is not irreducible. I tried the character table of $S_4$ and $D_4$ and thought about $\Bbb Z/6\Bbb Z$ but I couldn't find such ...
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### Quaternion group of order 12 [on hold]

What are the elements of the quaternion group of order 12 ? I found this on wikipedia but I don't know what are their elements.
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### Equivalence between these tensor product definitions

Let $V$ and $W$ be vector spaces. Then the tensor product $V \otimes W$ of $V$ and $W$ is the vector space $V \otimes W$ together with a bilinear map $\phi: V \times W \rightarrow V \otimes W$ such ...
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### Tower of Splitting Fields

If we have $F_n\geq\ldots\geq F_1$ fields such that $F_{i+1}$ is a separable splitting field over $F_i$ for all $i=1,\ldots,n-1$ is it true that $F_n$ is a splitting field over $F_1$? I think I can ...
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### A particular complex of integral group ring is exact: proof of Jacobson

Let $A$ be a $G$-module. Let $C_n=\mathbb{Z}G\otimes \cdots \otimes \mathbb{Z}G$ ($n+1$ copies). It is free $\mathbb{Z}$-module with basis $g_0\otimes g_1\otimes \cdots \otimes g_n$, $g_i\in G$. ...
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### On natural homomorphism $\nu:R\longrightarrow S^{-1}R$

Let $R$ be a commutative ring with $1_R$ and $S$ an multiplicatively closed set. We define the natural homomorphism \begin{align*} \nu:R &\longrightarrow S^{-1}R, \\ a&\longmapsto \nu(a):=\...
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