Linear representation theory of groups of order 2^n

This article describes the linear representation theory of groups of order 2^n, i.e., groups whose order is a power of $$2$$.

Important ways in which 2 differs from other primes
There are two very important differences between 2 and other primes:


 * Odd-order and ambivalent implies trivial, so a nontrivial finite $$p$$-group for odd $$p$$ cannot be ambivalent, i.e., not all its characters are real-valued. In particular, this means that the representations cannot all be realized over the field of real numbers, and in particular, there are no nontrivial examples of rational-representation groups or rational groups for odd primes. However, for the prime $$p = 2$$, there are many examples of rational-representation groups (such as elementary abelian 2-groups and dihedral group:D8), examples of rational groups that are not rational-representation groups (such as quaternion group), and examples of other ambivalent groups (such as dihedral group:D16).
 * Odd-order p-group implies every irreducible representation has Schur index one: This means that for an odd-order $$p$$-group, every irreducible representation can be realized over the field generated by its character values (Note: There do exist non-nilpotent odd-order groups with representations having Schur index values more than 1). This is not the case for 2-groups, and there exist irreducible representations of 2-groups with Schur index greater than 1. The smallest example is faithful irreducible representation of quaternion group, which has Schur index 2.
 * The multiplicative group of $$\mathbb{Z}/p^n\mathbb{Z}$$ is cyclic for odd $$p$$, but is not cyclic for $$p = 2, n \ge 3$$. Thus, we can construct examples of finite 2-groups such that the Galois group of a minimal cyclotomic splitting field over $$\mathbb{Q}$$ is not cyclic, and this is not possible for odd $$p$$. Therefore, 2 is the only prime where it is possible to construct examples where the orbit structure on the irreducible representations differs from the orbit structure on the conjugacy classes under the action of a Galois group. (Examples, links need to be provided).