Linked Questions
10 questions linked to/from Does every non-empty set admit a group structure (in ZF)?
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Is it provable in $\mathsf{ZF}$ that there is a group structure on any set $X$? [duplicate]
Given a set $X$ is it provable in $\mathsf{ZF}$ that there is a binary operation $\ast: X\times X\to X$ such that $(X,\ast)$ is a group?
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What are the most elegant proofs that you have learned from MO?
One of the things that MO does best is provide clear, concise
answers to specific mathematical questions. I have picked up ideas
from areas of mathematics I normally wouldn't touch, simply because
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How many morphisms from 1 to 1+1 can there be?
Here is an interesting question raised by Alice Rhyl.
Let $C$ be a category with a terminal object $1$ and finite coproducts. How many different morphisms $f : 1 \to 1 + 1$ can there be?
There are ...
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answers
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Making N (set of all positive integers) a group [closed]
Can anybody please give me an example of a binary operation under which N forms a group? More generally, how to find some operations to make possibly any set a group?
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answers
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Is Lagrange's Theorem equivalent to AC?
Lagrange's Theorem is most often stated for finite groups, but it has a natural formation for infinite groups too: if $G$ is a group and $H$ a subgroup of $G$, then $|G| = |G:H| \times |H|$.
If we ...
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answer
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Does every set admit a ring structure or a field structure?
Attempt to answer: Every set up to cardinality admits a ring structure, (except the empty set). Let $S$ be a set. If $S$ if finite, we may assume it's $ \mathbb{Z} / n \mathbb{Z}$. Otherwise, ...
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Group & modules of arbitrary cardinality [closed]
How do I see that there is a group of an arbitrary cardinality? Is this also true for abelian groups? Also, given a commutative ring $R\neq 0$ how do I see that there is an $R$-module of arbitrary ...
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Does every non-empty set admit an (affine) scheme structure (in ZFC)?
This question is partially inspired by this question: Does every non-empty set admit a group structure (in ZF)?
It was also inspired by my desire to explain the importance of quotient morphisms when ...
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A model with $\kappa$ many automorphism and a rigid element.
The following should be known, but I could not find an example.
Let $\kappa$ be an uncountable cardinal. Find a model $M$ of size $\kappa$ which has $\ge\kappa$ many automorphisms, but for some $m\in ...
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If every definable class admits a group structure, must global choice hold?
It is a remarkable fact, due to Hajnal and Kertész and explained very well in this MathOverflow answer by user Ashutosh, that the axiom of choice is equivalent to the assertion that every nonempty set ...
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