All Questions
Tagged with quotient-group notation
12 questions
2
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1
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97
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In $(G_1\times G_2)/G_2$, I am confused since $G_2$ is clearly not a subgroup of $G_1\times G_2$
I have seen the following expression in the text book of algebra chapter$0$.
$(G_1\times G_2)/G_2$. I am confused since $G_2$ is clearly not a subgroup of $G_1\times G_2$, and hence not a normal ...
- 21
-1
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1
answer
76
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How do I write an element of a quotient of a set of vectors by matrix in $GL_2(\Bbb Q)$? [closed]
My first time writing the quotient of a set of vectors by a matrix. Let the set of vectors $X=\pmatrix{x\\1}$ have $x\in\Bbb Q_2$ drawn from the 2-adic numbers.
Now let the matrices of the form $G=\...
- 9,436
0
votes
0
answers
106
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Notation: What is $C(G)$?
$G$ is the order $55$ group described by $G=\langle x,y \mid x^{11}=y^5=1, yxy^{-1}=x^4\rangle$. I am tasked with showing $|G/C(G)|=5$, but I don't know what $C(G)$ is supposed to be. It can't be the ...
1
vote
1
answer
123
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Arbitrary subset of quotient group or ring
Let $R$ be a group/ring and $I$ a normal subgroup/ideal, and form the quotient group/ring $R/I$.
Is is legitimate to write either of the following?
$S/I$ is an arbitrary subset of $R/I$, where $S \...
- 483
3
votes
2
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267
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Explanation with Examples: Finite Subgroup of Rational Numbers Modulo Squares and Its Elements
Let $\mathbb Q^{\times}/\mathbb Q^{\times^2}$ denote the group of rational numbers modulo squares. This
means that we regard two nonzero rational numbers $x_1, x_2$ as equivalent if the ratio $x_1/x_2$...
1
vote
1
answer
90
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Which notation is best for $R/I$
When $R$ is a ring and $I$ is an ideal of $R$, I have seen a variety of notational uses for the cosets in $R/I$, and I'm not sure which one is best in which context. For $a\in R$, if $C_a\in R/I$ is ...
- 3,201
1
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1
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25
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How are progressive quotients written?
Let $W=X/Y$ be a quotient of $X$
Let $W/Z$ be a quotient of $W$
To write $W/Z$ without $W$, Would we need to write $(X/Y)/Z$? I presume $X/Y/Z$ is ambiguous?
I'm imagining a group quotient and not ...
- 9,436
1
vote
1
answer
587
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What does the multiplication in cosets mean?
I was getting ready to learn about order of an element, cosets, and langrange's theorem in group theory. Consequently this involves multiplying two elements of a group. After seeing several examples ...
- 1,857
0
votes
1
answer
168
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Multiplicative quotient - what's the correct notation? Does a quotient group contain singletons or cosets?
I want to learn the correct notation and language with which to communicate clearly about a quotient on the multiplicative group $G=(\Bbb Q^+,\times)$
In particular, I want to know the notation with ...
- 9,436
0
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0
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29
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How do I write $\Bbb Z{\left[\frac16\right]}\cap\left[\frac23,\frac43\right),+$ modulo $\frac23$?
How do I talk about the elements of $\Bbb Z{\left[\frac16\right]}$ in the interval $\left[\frac23,\frac43\right)$ as a set with addition reduced by $\frac23$?
It would seem to be a torsion group with ...
- 9,436
2
votes
0
answers
170
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Notation for quotient space obtained by collapsing a subset to a point?
If $X$ is a topological space and $A$ is a (closed, usually) subset of $X$, then the quotient space obtained by "collapsing $A$ to a point" is often denoted by $X / A$.
Unfortunately, that notation ...
- 803
0
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1
answer
75
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Notational confusion about HNN-extensions: $G=K \ast_{H,t}$.
Let $G=K \ast_{H,t}$ denote an HNN-extension, i.e., $$H \le K \le G, H^t \le K.$$
Is it true that $\{K, t\}$ is a generating system for $G$? In particular, $G/K$ is cyclic?
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