Last revised: January 20, 2023 @ 11:39:05
The First Approach — Via Substructure Isomorphisms —
From "Symmetry in Mathematics and Mathematics of Symmetry"
by Peter J. Cameron, a Jan. 16, 2007, talk at the International
Symmetry Conference, Edinburgh, Jan. 14-17, 2007 —
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Local or global?
"Among other (mostly more vague) definitions of symmetry, the dictionary will typically list two, something like this:
• exact correspondence of parts;
• remaining unchanged by transformation.
Mathematicians typically consider the second, global, notion, but what about the first, local, notion, and what is the relationship between them? A structure M is homogeneous * if every isomorphism between finite substructures of M can be extended to an automorphism of M ; in other words, 'any local symmetry is global.' "
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A related discussion of the same approach —
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"The aim of this thesis is to classify certain structures
which are, from a certain point of view,
as homogeneous as possible, that is
which have as many symmetries as possible.
… the basic idea is the following: a structure S is
said to be homogeneous if, whenever two (finite)
substructures S1 and S2 of S are isomorphic,
there is an automorphism of S mapping S1 onto S2.”
— Alice Devillers,
“Classification of Some Homogeneous
and Ultrahomogeneous Structures,”
Ph.D. thesis, Université Libre de Bruxelles,
academic year 2001-2002
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The Wikipedia article Homogeneous graph discusses the local-global approach
used by Cameron and by Devillers.
For some historical background on this approach
via substructure isomorphisms, see a former student of Cameron:
Dugald Macpherson, "A survey of homogeneous structures,"
Discrete Mathematics , Volume 311, Issue 15, 2011,
Pages 1599-1634.
Related material:
Cherlin, G. (2000). "Sporadic Homogeneous Structures."
In: Gelfand, I.M., Retakh, V.S. (eds)
The Gelfand Mathematical Seminars, 1996–1999.
Gelfand Mathematical Seminars. Birkhäuser, Boston, MA.
https://doi.org/10.1007/978-1-4612-1340-6_2
and, more recently,
Gill et al., "Cherlin's conjecture on finite primitive binary
permutation groups," https://arxiv.org/abs/2106.05154v2
(Submitted on 9 Jun 2021, last revised 9 Jul 2021)
This approach seems to be a rather deep rabbit hole.
The Second Approach — Via Induced Group Actions —
My own interest in local-global symmetry is of a quite different sort.
See properties of the two patterns illustrated in a note of 24 December 1981 —
Pattern A above actually has as few symmetries as possible
(under the actions described in the diamond theorem ), but it
does enjoy, as does patttern B, the local-global property that
a group acting in the same way locally on each part induces
a global group action on the whole .
* For some historical background on the term "homogeneous,"
see the Wikipedia article Homogeneous space.
