# Tagged Questions

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### On direct sum and direct product of groups

I understood the definitions of direct sums and direct products of groups (rings, modules, etc). The definitions of them can be given in terms of universal properties - certain commutative diagram in ...
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### Are Free Groups the “Smallest Group” Containing their Generators

I apologize if this is a duplicate; I was not sure how to search for this. When I say "the smallest group" I mean unique up to isomorphism of course. Specifically, is "the smallest group containing ...
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### How to prove pairwise independence of a family of hash functions?

I want to prove pairwise independence of a family of hash functions, but I don't know where to start. Given the family of hash functions: H with h(x) = a * x + b (mod M). ( Say h: U -> V, then: M ...
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### Proof of the universal property of the quotient topology

In this question: universal property in quotient topology I saw the following theorem: Let $X$ be a topological space and $\sim$ an equivalence relation on $X$. Let $\pi: X\to X/{\sim}$ be the ...
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### Homomorphisms between two universal free groups [duplicate]

I'm trying to prove that, for positive integers, $m$ and $n$, there is a homomorphism from $F_n$ onto $F_m$ if and only if $m$ is less than or equal to $n$ (where $F_n$ is the universal free group ...
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### Does every continuous map between compact metrizable spaces lift to the Cantor set?

I'm interested in the universal properties of the Cantor set. It is well-know that the Cantor set $2^\mathbb{N}$ is "universal" in the category of metrizable compact spaces, in the sense that every ...
### What is the class of topological spaces $X$ such that the functors $\times X:\mathbf{Top}\to\mathbf{Top}$ have right adjoints?
For any topological space $X$, define a functor $\times X:\mathbf{Top}\to\mathbf{Top}$ by $Y\mapsto Y\times X$ (and acting on the hom-sets in the natural way). I know that if $X$ is locally compact, ...