# Tagged Questions

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### Notation for pointwise versus “setwise” stabilizers

Suppose one is working with both pointwise and setwise stabilizers of sets under a group action. Are there common conventions for notationally distinguishing these two notions? How common are they? ...
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### What does “$\cong$” sign represent?

I came across this sign when reading some papers. I looked up Wikipedia. It says "The symbol "$\cong$" is often used to indicate isomorphic algebraic structures or congruent geometric figures." So ...
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### Notation Question for summations

I came across this notation in a textbook of Algebra. With respect to the definition of linear independence in a Vector Space $V$. We define a subset $S = \{\alpha_i \ | \ i \in I\} \subset V$ as ...
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### Notation for quotient ring

I'm wondering where the notation for the quotient of a ring by an ideal comes from. I.e., why do we write $R/I$ to denote a ring structure on the set $\{r+I: r\in I\}$, wouldn't $R+I$ be more ...
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### why notate coset representatives as $a h_1$ to show well-defined-ness?

In Fraleigh, there is what appears to be a classic theorem on cosets. I'm confused about the proof for the converse. Theorem. Let $H$ be a subgroup of a group $G$. Then left coset multiplication ...
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### Definition of a field

I have it defined here that a field $\mathbb{F}$ consists of $\mathbb{F} = (\mathbb{F},+,0,1,\cdot)$ where $\mathbb{F}$ is a set $0,1 \in \mathbb{F}$, $0 \not = 1$ $+$ is a mapping (addition), ...
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### What does $K^{1/p}$ for a field $K$ mean?

In the proof of the finite generation of the invariant ring of a finite group acting on $k[x_1,\dots,x_n]$, at one time there is a symbol I don't understand. The situation is as follows. $k$ is a ...
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### Looking for standard and consistent notation/terminology for the finite sequences/heaps on a set

Question 1. Does anyone know of standard and consistent notation for the following? The set of non-empty finite sequences on a set. As above, but including the empty sequence. The set of finite ...
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### Any commutative associative operation can be extended to a function on nonempty finite sets

This is a fact we use very frequently in general mathematics when we write such notations as $1+2+3+4$: since we know that $+$ is commutative and associative, we can just "drop the parentheses" and ...
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### What is the meaning of the parentheses in $\phi^{-1}\left[\{\phi(g)\}\right]=gH=Hg$?

I am studying homomorphisms is groups and i saw a theorem saying: For $g$ in a group $G$, the cosets $gH$ and $Hg$ are the same, and collapsed onto the single element $\phi(g)$ by $\phi$. That is, ...
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I am confused about the notation $\operatorname{rad}^2 A$. It can be considered as $\operatorname{rad}(\operatorname{rad}(A))$ or as $(\operatorname{rad}(A))^2$. Are ...
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### Some questions regarding the convention used

I've some questions regarding the following problem from Herstein. BTW I'm not looking for its solution: Do $\lambda_g$ is actually $\lambda_g(x)=xg$ when I write $x\lambda_g$ as $\lambda_g(x)?$ ...
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### Notation $f^{-1}$ for the inverse vs. notation $f^n$ for $n$-fold application

We often use to denote invertible function of $f$, as $f^{-1}$. In applied mathematics, This is the general rule. But very rigorous concepts of mathematics, like axiomatic number systems, it is just a ...
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### $H ≤G$ means $H$ is a subgroup of $G$?

I was reading this page: http://www.proofwiki.org/wiki/Definition:Subgroup I never heard that $H ≤G$ means $H$ is a subgroup of $G$. Is this standard notation ? And if not, what is/are normal ...
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### Is $A \times B$ the same as $A \oplus B$?

When $A, B$ are $K$-modules, then $A \times B$ is the same as $A \oplus B$. Let $A, B$ be two $K$-algebras, where $K$ is a field. Is $A \times B$ the same as $A \oplus B$? Thank you very much. ...
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### Is there a rigorous theory of context, whereby sets can gain additional structure within a context?

Consider sets $G$ and $H$ and a function $f : G \rightarrow H$. So far, it doesn't really make sense to ask whether $G$ and $H$ are groups (technically, the answer is "no, they're not groups"), and ...
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### Put the $i$th term to be the coefficient of $x^i$: A question about Mathematical notation and sequences

Take $u(x)$ and $v(x)$ to be integer polynomials, and then interpret them as sequences in the obvious way: i.e. you put the $i$th term to be the coefficient of $x^i$. $u(x)=x-2$, $v(x)=3x^2+x$. The ...
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### Notations for one-sided coset spaces

Let $G$ be a group and let $H$ be its subgroup. What notations are used for the coset spaces $\{gH\,|\,g\in G\}$ and $\{Hg\,|\,g\in G\}?$ What I've written is a bit much to type, and I will be writing ...
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### What does $R_P$ mean, for a ring $R$ and an ideal $P$?

What does $R_P$ mean, for a ring $R$ and an ideal $P$? This appeared in some notes by a teacher of mine, but he didn't define this notation. He used it as follows: suppose $R$ is a commutative ring, ...
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### What's a good notation for an endomorphism induced on the quotient?

For example, suppose that $V$ is a vector space, $A \in \operatorname{End}V$, and $U \subset V$ is an $A$-invariant subspace, and $\pi_{V/U}: V \to V/U$ the natural projection. Then $A$ induces a ...
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### Interpretation of Symbol: “$\rtimes$”

What exactly does $\text{Aff}(2) = \mathbb{R}^2 \rtimes SL_{2}(\mathbb{R})$ mean? I know that it is the group of area preserving affine transformations of (oriented) $\mathbb{R}^2$. But how would you ...
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### Original papers on the subject of group actions

Does anyone if there are any original paper(s) that first introduced the notion of group action or permutation representation, and who the author(s) were? Any references I have found so far on e.g. ...
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### Notations in Group theory

I will start by apologizing as many will not like this question. I am reading the paper COHOMOLOGY THEORY OF GROUPS WITH A SINGLE DEFINING RELATION and having focused on typology throughout my ...
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### What does $s^t$ mean in group theory?

For subset $S$ and $T$ of a group, define $ST = \{st|s \in S, t \in T\}$ and $S^T = \{s^t|s \in S, t \in T\}$. What does $s^t$ mean in this context?
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### Notation for n-ary exponentiation

We have $n$-ary sums ($\sum$) and products ($\prod$). Is there an $n$-ary exponentiation operator? \underset{i=1}{\overset{n}{\LARGE{\text{E}}}}\, x_i = x_1 \text{^} (x_2 \text{^} (\cdots \text{^} ...
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### On 'backslash-forward slash' notation

I am curious about a notation that I have seen, but I have only seen it in contexts beyond my current level of ability and so haven't learned its meaning. Also, it's often difficult to search for the ...
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### Polynomial ring as a ring of functions

My book describes the polynomial ring $R[X]$ as: $R[X] = R[\mathbb{N}] = \{f: \mathbb{N} \rightarrow R | \hspace{ 2mm} f(n) = 0, n \gg 0\}$. What is exactly meant by this? What do the double arrows ...
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### Why do mathematicians use this symbol $\mathbb R$ to represent the real numbers?

So, I'm wondering why mathematicians use the symbols like $\mathbb R$, $\mathbb Z$, etc... to represent the real and integers number for instance. I thought that's because these sets are a kind of ...
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### Notation for modules.

Given a module $A$ for a group $G$, and a subgroup $P\leq G$ with unipotent radical $U$, I have encountered the notation $[A,U]$ in a paper. Is this a standard module-theoretic notation, and if so, ...
given two elements $r$,$s$ in a ring $R$, are the following two notations equivalent? $(r,s)$ $(r)+(s)$ For example, in the ring $\mathbb{Z}[X]$, is $(2,X)=(2)+(X)$? Thanks a lot.
### Field of fractions of $R[X]$
Let $R$ be a domain and let $Q$ be its field of fractions. Show that the field of fractions of $R[X]$ is isomorphic to $Q(X)$. By the way, I don't know exactly what $Q(X)$ is. It means $Q[X]$? Or ...