2-group

In mathematics, a 2-group is a group in which every element has order a power of 2. Equivalently, a 2-group is a group whose order is a power of 2, when the group is finite. These are also sometimes called dyadic groups.

Definition

A group G is a 2-group if, for every g in G, there exists a non-negative integer n such that g²ⁿ = e, where e is the identity element of G. For finite groups, this is equivalent to saying that the order of the group, |G|, is a power of 2, i.e., |G| = 2^k for some non-negative integer k. For infinite groups, the condition is that every element has order a power of 2, but the group's cardinality is not necessarily a power of 2.

Properties

• Every subgroup and every quotient group of a 2-group is itself a 2-group.
• The Sylow 2-subgroups of any finite group are 2-groups. Indeed, the Sylow 2-subgroups are maximal 2-subgroups.
• Finite 2-groups are nilpotent. In fact, a finite group is nilpotent if and only if it is the direct product of its Sylow subgroups.
• The center of a nontrivial finite 2-group is nontrivial. That is, if G is a 2-group with more than one element, then the center Z(G) is not equal to {e}.
• The Burnside Basis Theorem states that if G is a finite 2-group, the minimal number of generators of G is equal to the dimension of the vector space G/Φ(G) over the field with two elements, where Φ(G) is the Frattini subgroup of G.

Examples

• The cyclic group of order 2ⁿ, denoted by C_2ⁿ or Z/(2ⁿZ), is a 2-group.
• The Klein four-group V₄, which is isomorphic to Z/2Z × Z/2Z, is a 2-group.
• The dihedral group of order 2ⁿ for n ≥ 3 is a 2-group.
• The quaternion group Q₈ is a 2-group.
• The infinite quasicyclic group, denoted Z(2^∞), is a 2-group. It consists of all complex numbers of the form e^2πi(a/2n) for some non-negative integer n and integer a, under complex multiplication.

Related Concepts

• p-group: A generalization of 2-groups to groups where the order of every element (or the order of the group in the finite case) is a power of a prime number p.
• Sylow theorems: Theorems concerning the existence and properties of maximal p-subgroups (Sylow p-subgroups) of a finite group.