A local group is a T2-space 
with an identity element 
and continuous
multiplication and inversion on possibly restricted domains (Goldbring 2010, pp. 1270-1271).
Multiplication is defined on an open set 
containing every pair 
and 
, and inversion is defined on an open
neighborhood 
of 
. The identity law is 
. For 
, the products 
and 
are defined and equal 
. Local associativity requires
whenever 
, 
,
, and 
are all defined.
Restricting a topological group to an open neighborhood of its identity element, retaining
only products and inverses whose values remain in that neighborhood,
gives a local group. For example, the open interval 
has a local group structure under
addition, with 
defined only when 
. Thus 
can be added to 
, but not to 
within this local group. A local
isomorphism between two local groups is represented by a homeomorphism
between open neighborhoods of their identity
elements that carries identity to identity and preserves the restricted products
and inverses in both directions (Goldbring 2010, p. 1274).
A local group is locally Euclidean when its identity element has an open
neighborhood homeomorphic to an open
set in 
for some nonnegative integer 
. Goldbring (2010, p. 1269) proved that every locally
Euclidean local group admits a local isomorphism
to a Lie group, settling the local-group version of
the fifth of Hilbert's problems.
This notion is distinct from local group theory, which studies finite groups through their local subgroups. In that setting, a local subgroup is the normalizer of a nontrivial subgroup of prime power group order (Aschbacher 2004, p. 738). It is an actual subgroup of the ambient finite group, so its products and inverses are always defined. The term local group instead permits restricted domains for these operations and does not require an ambient group.
