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Can we ascertain that there exists an epimorphism $G\rightarrow H$?

Let $G,H$ be finite groups. Suppose we have an epimorphism $$G\times G\rightarrow H\times H$$ Can we find an epimorphism $G\rightarrow H$?
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Connected metric spaces with disjoint open balls

Let $X$ be the $S^1$ or a connected subset thereof, endowed with the standard metric. Then every open set $U\subseteq X$ is a disjoint union of open arcs, hence a disjoint union of open balls. Are ...
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Is there a characterization of groups with the property $\exists N \unlhd G : \not \exists H \leq G \,\,\,s.t.\,\,\,H\cong G/N$?

A common mistake for beginning group theory students is the belief that a quotient of a group $G$ is necessarily isomorphic to a subgroup of $G$. Is there a characterization for groups for which this ...
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How to think of the group ring as a Hopf algebra?

Given a finite group $G$ and a field $K$, one can form the group ring $K[G]$ as the free vector space on $G$ with the obvious multiplication. This is very useful when studying the representation ...
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Unexpected approximations which have led to important mathematical discoveries

One often finds at MSE approximate numerology questions like Prove $\log_{\frac{1}{4}} \frac{8}{7}> \log_{\frac{1}{5}} \frac{5}{4}$, Prove $(\dfrac{2}{5})^{\frac{2}{5}}<\ln{2}$, Comparing ...
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Can one deduce Liouville's theorem (in complex analysis) from the non-emptiness of spectra in complex Banach algebras?

As you probably know, the classical proof of the non-emptiness of the spectrum for an element $x$ in a general Banach algebra over $\mathbb{C}$ can be proven quite easily using Liouville's theorem in ...
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All polynomials with no natural roots and integer coefficients such that $\phi(n)|\phi(P(n))$

Let $P(x)$ be a polynomial with integer coefficients such that the equation $P(x)=0$ has no positive integer solutions. Find all polynomials $P(x)$ such that for all positive integers $n$ we have ...
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Does a four-variable analog of the Hall-Witt identity exist?

Lately I have been thinking about commutator formulas, sparked by rereading the following paragraph in Isaacs (p.125): An amazing commutator formula is the Hall-Witt identity: ...
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Number of simple edge-disjoint paths needed to cover a planar graph

Let $G=(V,E)$ be a graph with $|E|=m$ of a graph class $\mathcal{G}$. A path-cover $\mathcal{P}=\{P_1,\ldots,P_k\}$ is a partition of $E$ into edge-disjoint simple paths. The size of the cover is ...
Closed form for $\int_0^\infty\log\frac{J_\mu(x)^2+Y_\mu(x)^2}{J_\nu(x)^2+Y_\nu(x)^2}\mathrm dx$
Consider the following integral: $$\mathcal{I}(\mu,\nu)=\int_0^\infty\log\frac{J_\mu(x)^2+Y_\mu(x)^2}{J_\nu(x)^2+Y_\nu(x)^2}\mathrm dx,$$ where $J_\mu(x)$ is the Bessel function of the first kind: ...