Answer to Question #23880 in Abstract Algebra for Irvin

Question

Let G be the group of order 21 generated by two elements a, b with the relations a^7 = 1, b^3 = 1, and bab^−1 = a^2. How about RG and the real representations of G?

Answer 1

Tensoring QG ∼ Q × Q(ω) ×M₃(Q(√−7)),up to R, we get RG ∼ R × C ×M₃(C),which means that each of the irreducible Q-representations above remainsirreducible over R (and these give all irreducible R-representations). We see
that the 6-dimensional irreducible
rational (resp. real) representationof G tensors up into the direct sum of the two 3-dimensional complexrepresentations of G. (Of course, this is also easy to see by checkingdirectly that χK= χ4 + χ5.)