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Local Group Presentation

A local group presentation records only those products of generators that remain in a given generating set. For a group G, let X be an inverse-closed generating set containing the identity. Its universal completion is the group presentation

U(X)=<[x] (x in X)|[1]=1, [x][y]=[xy] when x,y,xy in X>.

The map [x]|->x gives a canonical surjective group homomorphism U(X)->G (Pan 2026).

Pan (2026) reported that this map is an isomorphism for every such generating set X if and only if G is the trivial group, the cyclic group C_2, or the cyclic group C_3, resolving a conjecture of P. A. Smith. This notion is distinct from finite local group theory. Pan (2026) reports using ChatGPT to explore, develop, and check arguments, followed by author revision.

See also

Group Presentation

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References

Pan, Y. "Local Presentations of Groups: Completion and Smith's Conjecture." 6 Oct 2026. https://arxiv.org/abs/2610.10586.

Cite this as:

Weisstein, Eric W. "Local Group Presentation." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/LocalGroupPresentation.html

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