Rational not implies rational-representation

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This article gives the statement and possibly, proof, of a non-implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., rational group) need not satisfy the second group property (i.e., rational-representation group)
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Statement

It is possible to have a group (specifically, a finite group) that is a rational group (in the sense that all its character values are rational) but is not a rational-representation group (in the sense that not all its irreducible representations over the complex numbers can be realized over the rationals).

Such an example group must have at least one irreducible linear representation with Schur index greater than 1.

Proof

Example of the quaternion group

Further information: quaternion group, linear representation theory of quaternion group

The quaternion group of order eight is an example. It has a two-dimensional irreducible representation over the complex numbers with a rational character, but which cannot be realized over the rationals. The double of this representation can be realized over the rationals.