There is an ubiquitous pattern of questions concerning probably any kind of mathematical object or structure: groups, graphs, numbers, categories, and so on. It goes like this (informally):
> Can a class of objects or structures of a given kind *X* that is characterized by some "external condition" *Y* be defined by a condition *Z* in their respective "internal"
> language, and if so: how?
Well-known examples ("external condition" **=** "internal condition"):
1. groups $G$ isomorphic to a subgroup of the symmetric group on $G$ **=** all groups (*Cayley's theorem*)
2. graphs embeddable in the plane **=** graphs not containing a subgraph that is a subdivision of $K_5$ or $K_{3,3}$ (*Kuratowski's theorem*)
3. numbers *n* of trees on labeled vertices **=** numbers *n = k*<sup>*k*-2</sup> for some *k* > 1 (*Cayley's theorem on trees*)
4. numbers *n* with only one group of order *n* **=** numbers *n = p<sub>1</sub> · p<sub>2</sub> · ... · p<sub>k</sub>* for some *k* > 0, where the *p<sub>i</sub>* are distinct primes and no *p<sub>j-1</sub>* is divisible by any *p<sub>i</sub>* (*cyclic numbers*, see [Sloane's A003277][1])
Further examples from MO:
- [Which graphs are Cayley graphs?][3]
- [Can we recognize when a category is
equivalent to the category of models
of a first order theory?][4]
- [Can you determine whether a graph is
the 1-skeleton of a polytope?][5]
> **Question #1:** What's the proper way to
> characterize this pattern of
> questions? What's the common
> context / rationale?
***
> **Question #2:** How is the introductory question to be
> posed properly?
[1]: http://www.research.att.com/~njas/sequences/A003277
[2]: http://www.research.att.com/~njas/sequences/A003277
[3]: http://mathoverflow.net/questions/14830/which-graphs-are-cayley-graphs
[4]: http://mathoverflow.net/questions/13155/can-we-recognize-when-a-category-is-equivalent-to-the-category-of-models-of-a-fir
[5]: http://mathoverflow.net/questions/9255/can-you-determine-whether-a-graph-is-the-1-skeleton-of-a-polytope