A local isomorphism between local groups and is represented by a homeomorphism between open neighborhoods of their identity elements. Both and must preserve the identity element. A restricted product or inverse is defined on either side iff the corresponding product or inverse is defined on the other, and its value must be preserved by the maps (Goldbring 2010, p. 1274). Representatives that agree on a smaller open neighborhood of the identity element represent the same local isomorphism. The local groups are called locally isomorphic when such a local isomorphism exists.

Even for topological groups, local isomorphism does not require an isomorphism of the full groups. For example, the additive real line and the unit circle under complex multiplication are locally isomorphic via the map on sufficiently small open neighborhoods of the identity elements, although the full groups are not isomorphic.

See also

Isomorphism, Local Group, Topological Group

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References

Goldbring, I. "Hilbert's Fifth Problem for Local Groups." Ann. Math. 172, 1269-1314, 2010. https://doi.org/10.4007/annals.2010.172.1269.

Cite this as:

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

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