All Questions
Tagged with cardinals abstract-algebra
46
questions
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For $k$-algebras $B_1, \dots, B_n$, $\# \operatorname{Hom}_k( \prod_{i=1}^nB_i, \Omega) = \Sigma_{i=1}^n \# \operatorname{Hom}_k(B_i, \Omega)$?
Let $k$ be a field with $ k \subseteq \Omega$ a algebraically closed field. Let $B_1 , \dots, B_n$ be ( possibly finite local ) $k$-algebras. Then next equality of cardinals holds
$$ \# \operatorname{...
- 2,656
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Can a countable union of subgroups of uncountable index in G be equal to G? [closed]
Let G be a group and $\{H_i\}_{i<\omega}$ be a countable family of subgroups of $G$, each of them of uncountable index. Can $G=\bigcup_{i<\omega} H_i$?
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Group action & cardinality of a set.
You can find here more details and explanation on this question.
Question:
Let $n$ be a non-negative integer. For any family $ (i_1, \ldots, i_r) $ of non-negative integers such that $ i_1 + \ldots + ...
- 719
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2
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106
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Torsion-free commutative groups of a given cardinality
If an abelian group $G$ is torsion-free, it has at least one subgroup isomorphic to $\Bbb{Z}$, given by $⟨g⟩ := \{ g^n | n \in \Bbb{Z} \}$, for any non-identity element $g$. This subgroup is obviously ...
- 5,830
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Theorem 5, Section 1.4 of Hungerford’s Abstract Algebra
If $K,H,G$ are groups with $K\lt H \lt G$, then $[G:K]=[G:H][H:K]$. If any two of these indices are finite, then so is the third.
Proof: By Corollary 4.3 $G= \bigcup_{i\in I}Ha_i$ with $a_i \in G$, $|...
- 4,249
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Uncountability of the set of functions from the set of natural numbers to itself
I'm new to this website so forgive me for any formatting mistakes or other errors you encounter. I would appreciate it if you pointed them out. For anyone interested, the problem is from Algebra by ...
- 11
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The cardinality of the maximal linearly independent susbet of a free module over a commutative ring with the identity
Let $R$ be a commutative ring with the identity and $F$ be a free-module over $R$.
I assume $X$ is the basis of $F$.I know any two bases of $F$ have the same cardinality.
My question is that can we ...
- 1,159
2
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2
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136
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Do hypercontinuous fields exist?
"Hypercontinuity" is a cardinality of a continuous set's power set (set of all subsets). When talking about fields, I mean the cardinality of field's set. At first glance there is nothing ...
- 21
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Degree of a field extension by a transcendental element
Let $F$ be a field, and let $F(x)$ be the field of fractions of the polynomial ring $F[x]$. I'm interested in the degree of the field extension $[F(x) : F]$. Obviously it is infinite, but what exactly ...
- 6,610
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Cardinality of double cosets
We show that the sets of double cosets $K\backslash G/H$ is in bijection with $K\backslash(G/H)$. So $\vert K\backslash(G/H) \vert = \vert G/H \vert /\vert K \vert = (\vert G \vert / \vert H \vert)/\...
- 4,256
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56
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Ratio of two infinite cardinal numbers
Suppose $G$ is the group of all functions between $[0,1]\to\mathbb{Z}$. Let $H$ be the subgroup defined as $H=\{f\in G: f(0)=0\}$. Then, what can be said about the cardinality of $H$ and its index in $...
- 7,085
3
votes
2
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Number of elements in a Group such that $x^7=e$
Given that $G$ is a finite Group. Prove that number of elements in G such that $x^7=e$ where $x \in G$ is always Odd.
My attempt:
First possibility is a Trivial group since $e^7=e$. Trivial Group ...
- 10.2k
15
votes
1
answer
382
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Must large (infinite) groups have large automorphism groups?
For every cardinal $\kappa$, is there a cardinal $\lambda$ such that for all groups $G$ with $|G| > \lambda$, we have $|\mathrm{Aut}(G)| > \kappa$? I believe a similar result holds for finite ...
- 2,636
3
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How to decide the cardinality of $\{\text{all group isomorphisms from }(\mathbb{R},+)\text{ onto }(\mathbb{R}^+,\cdot)\}$?
The additive group of reals $(\mathbb{R},+)$ and the multipilicative group of positive reals $(\mathbb{R}^+,\cdot)$ are isomorphic, and $x \mapsto \exp(x)$ is one isomorphism from $(\mathbb{R},+)$ ...
user695163
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Let $S$ be the set of all possible functions mapping $\{\sqrt 2, \sqrt 3, \sqrt 5, \sqrt 7 \}$ to $\Bbb Q$, find the cardinality of $S$.
Let $S$ be the set of all possible functions mapping $\{\sqrt 2, \sqrt 3, \sqrt 5, \sqrt 7 \}$ to $\Bbb Q$, find the cardinality of $S$.
At first I wanted to use the theorem that for any sets $A$ and ...
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