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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.

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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