Wikipedia
In mathematics, an n-group, or n-dimensional higher group, is a special kind of n-category that generalises the concept of group to higher-dimensional algebra. Here, n may be any natural number or infinity. The thesis of Alexander Grothendieck's student Hoàng Xuân Sính was an in-depth study of 2-groups under the monniker 'gr-category'.
The general definition of n-group is a matter of ongoing research. However, it is expected that every topological space will have a ''homotopy n-group'' at every point, which will encapsulate the Postnikov tower of the space up to the homotopy group π, or the entire Postnikov tower for n = ∞.
The definition and many properties of 2-groups are already known. A 1-group is simply a group, and the only 0-group is trivial. 2-groups can be described using crossed modules.
- n-group (category theory), an N-category that possesses a group-like structure.
- N-group (finite group theory), a finite group all of whose local subgroups are solvable.
In mathematical finite group theory, an N-group is a group all of whose local subgroups (that is, the normalizers of nontrivial p-subgroups) are solvable groups. The non-solvable ones were classified by Thompson during his work on finding all the minimal finite simple groups.