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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 {\displaystyle 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 moniker 'gr-category'.
The general definition of n {\displaystyle n} -group is a matter of ongoing research. However, it is expected that every topological space will have a homotopy n {\displaystyle n} -group at every point, which will encapsulate the Postnikov tower of the space up to the homotopy group π n {\displaystyle \pi _{n}} , or the entire Postnikov tower for n = ∞ {\displaystyle n=\infty }.