2group

2group, in the mathematical literature, most often refers to a 2-group: a finite group whose order is a power of 2. More precisely, a finite group G is a 2-group if |G| = 2^n for some n ≥ 0. In many texts the term extends to infinite groups by requiring that every element has order a power of 2, though the standard focus is on finite 2-groups. The concept is a special case of p-groups, which are finite groups of order p^n for a prime p, with p = 2.

Properties of finite 2-groups include foundational structure results. Every finite 2-group is solvable and nilpotent; in particular, the center Z(G) is nontrivial, a consequence of the class equation. These groups exhibit rich subgroup lattices and have many normal subgroups, reflecting their hierarchical composition via central and derived series. The Frattini subgroup often plays a key role in understanding their maximal subgroups.

Common examples range from simple to highly nonabelian. The cyclic groups C_{2^n} of order 2^n are abelian examples. Nonabelian examples include the dihedral groups D_{2^n} (order 2^n for n ≥ 2), the quaternion groups Q_8 and more generally generalized quaternion groups Q_{2^m} (m ≥ 3), and certain semidihedral and other 2-groups. There exist many nonisomorphic 2-groups of a given order, making their complete classification intricate beyond small orders.

In summary, 2groups occupy a central role in the study of finite p-groups, offering insight into group structure, subgroup behavior, and the broader landscape of algebraic groups. They are a key example of how prime-power order constrains algebraic properties.