No small subgroup

Short description: Restriction on topological groups in mathematics

In mathematics, especially in topology, a topological group [math]\displaystyle{ G }[/math] is said to have no small subgroup if there exists a neighborhood [math]\displaystyle{ U }[/math] of the identity that contains no nontrivial subgroup of [math]\displaystyle{ G. }[/math] An abbreviation '"NSS"' is sometimes used. A basic example of a topological group with no small subgroup is the general linear group over the complex numbers.

A locally compact, separable metric, locally connected group with no small subgroup is a Lie group. (cf. Hilbert's fifth problem.)

See also

• Hilbert's fifth problem § No small subgroups

References

• M. Goto, H., Yamabe, On some properties of locally compact groups with no small group