# Questions tagged [classification]

Classification of various mathematical structures. For classification in the sense of statistics / machine learning, use [tag:statistical-classification].

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### CFSG-free proof for classifying simple $K_3$-group

Let $G$ be a finite nonabelian simple group. We call $G$ a $K_3$-group if $|G|=p^aq^br^c$ where $p,q,r$ are distinct primes and $a,b,c$ are positive integers. My question is: Is there a CFSG-free ...
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### On classifying groups of order $p^5$

Can someone suggest me some source where the author has classified all non-isomorphic groups of order $p^5$ ? I need complete classification (not upto isoclinism), and also in finitely presented form ....
314 views

### Is there a precise relationship between the goals of moduli theory and the minimal model program?

I want to get into some of the big classification problems in algebraic geometry, but have a very broad question. Ultimately we would like to classify all varieties over some field up to isomorphism, ...
394 views

### Comonoids in the category of monoids

Let us give the category of monoids $\mathbf{Mon}$ a monoidal structure with $\otimes = \sqcup$ (coproduct). How can we classify $\mathbf{CoMon}(\mathbf{Mon})$, the category of comonoids of monoids? ...
58 views

### Matroids which are transitive on minimal basis exchanges

I am looking for matroids in which all minimal basis exchanges look the same, that is, the matroid is transitive on these. Let me explain what I mean by that. Consider a finite matroid $M$. Define a ...
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### Why the VC dimension of triangles in 2D space is not greater than 7?

I understand that there are sets of 7 points on a circle that can be fully shattered using triangles.But, it is not clear to me why it cannot shatter 8 points. Is there any intuitive way of arriving ...
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### Name for "partially complete" invariants in classification problems?

For any equivalence $\sim$ on some collection of objects $C$ consider the problem of trying to determine if two arbitrary objects $x$ and $y$ in $C$ are equivalent i.e. if $x\sim y$ now by definition ...
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### The "Johnson polychora"

Firstly, a definition: A convex polyhedron, whose faces are regular polygons (2D polytopes). This includes the 92 Johnson solids, 13 Archimedean solids, 5 Platonic solids and two infinite ...
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### Classification of the quotients of the ring Z/4 [X]

Is it possible to classify all cyclic $\mathbb{Z}/4$-algebras, i.e. the regular quotients of $\mathbb{Z}/4 [X]$? A typical example is $\mathbb{Z}/4 [X] / \langle X^n , 2 X^k \rangle$. For my purposes ...
1 vote
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### Generalize characterization of upper semicontinous functions

Let $X$ be a metric space and denote $f:X \rightarrow \mathbb{R}.$ It is easy to show that the following two statements are equivalent: $(1)$ For any real number $c$, we have $f^{-1}(-\infty,c)$ is ...
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### Classification of pointed Hopf algebras up to gauge equivalence

The classification of finite-dimensional pointed Hopf algebras over an algebraically closed field of characteristic zero and whose group of group-like elements is abelian is very much completed. ...
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### Classification of non-Hausdorff topological vector spaces

It is well-known that up to topological isomorphism there is exactly one Hausdorff topological vector space (say, over $\mathbb{C}$) of a given dimension $n$, namely $\mathbb{C}^n$ with the euclidean ...
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### Smallest number $n$ for which we don't know the classification of all groups of order $n$

I noticed that in groupprops and wikipedia there are often given tables of classifications of groups of small order. This motivated me to ask, what is the current state of research in classifying all ...
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### Regular homotopy of punctured surfaces

A theorem of James and Thomas (Note on the classification of cross-sections, Topology 4) asserts that the space of immersions, up to regular homotopy, from a compact surface $S$ into $\mathbb{R}^3$ ...
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### Has anyone catalogued the "first generation" proof of the classification of finite simple groups?

It has been estimated that the original proof of the CFSG spans around 15,000 journal pages written by hundreds of authors over most of the 20th century. The GLS project attempted to simplify this ...
1 vote
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### Classification of finite subgroup of $PGSp_4(\mathbb{C})$

Is there a classification of the finite subgroups of $PGSp_4(\mathbb{C})$?
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### Classification of cubic surfaces in $\mathbb{P}^3$

We know every cubic surface in $\mathbb{P}^3$ is obtained by blowing up $\mathbb{P}^2$ at 6 points in general position. Hence they are all birational to $\mathbb{P}^2$. My question is: Do we have ...