In mathematics, especially in

topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...

, a

topological group In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two st ...

G

is said to have no small subgroup if there exists a neighborhood

U

of the identity that contains no nontrivial subgroup of

G.

An abbreviation '"NSS"' is sometimes used. A basic example of a topological group with no small subgroup is the

general linear group In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, ...

over the complex numbers. A locally compact, separable metric,

locally connected In topology and other branches of mathematics, a topological space ''X'' is locally connected if every point admits a neighbourhood basis consisting entirely of open, connected sets. Background Throughout the history of topology, connectedness ...

group with no small subgroup is a Lie group. (cf.

Hilbert's fifth problem Hilbert's fifth problem is the fifth mathematical problem from the problem list publicized in 1900 by mathematician David Hilbert, and concerns the characterization of Lie groups. The theory of Lie groups describes continuous symmetry in mathem ...

References

* M. Goto, H., Yamabe
On some properties of locally compact groups with no small group
Group theory #05 Lie groups Topological groups {{math-hist-stub

See also

References