ABSTRACT
Let G be a finite group and QG the rational group algebra. It is well known by Maschke’s theorem that QG is a semisimple algebra, then in order to classify these algebras, we compute the Wedderburn decomposition of its rational group algebras. This decomposition became important because of its applications in the theory of group representations, and to determine the units group of ZG. For this classification we use the group table notation of Thomas and
Wood (see [8]). It is easy to prove that if G is decomposable, i.e., G = H1 ×H2, then QG = QH1⊗QH2, and if G = 〈a : an = 1〉 is a cyclic group, then
QG ' ⊕ d|n
Q[x] 〈ϕd(x)〉 '
⊕ d|n Q(ξd),
where ϕd(x) is a dth cyclotomic polynomial. Thus, we are only going to show the Wedderburn decomposition of the group algebras over non abelian indecomposable groups of order greater than or equal to 16 and less than or equal to 32. In general, this decomposition is known for some group families like per-
mutation, dihedral, semidihedral, and quaternion groups (see [1]). In fact, if G = Sn then QG =Mn1(Q)⊕Mn2(Q)⊕ · · · ⊕Mnk(Q) with
and k is the number of conjugacy classes, if G is the dihedral group of order 2n, i.e.,
G = D2n = 〈x, y : xn = 1, y2 = 1, xy = yx−1〉 191
then QG ∼=
where Ad ∼= Q⊕Q if d = 1, 2 and Ad ∼=M2(Q[ξd+ ξ−1d ]) if d > 2, if G is the semidihedral group of order 16n, i.e.,
G = D−16n = 〈x, y : x8n = 1, y2 = 1, xy = yx4n−1〉 then
QG ∼= D8n ⊕
M2(Q[ξd − ξ−1d ]),
where k and m are nonnegative integers with m odd, such that n = 2km, and if G is the quaternion group of order 4n, i.e.,
G = Q4n = 〈x, y : x2n = 1, y2 = xn, xy = yx−1〉 then
QG ∼= QDn ⊕ d=2kr r|m
Q[ξ2d, j],
where k and m are nonnegative integers with m odd, such that n = 2km, j2 = −1 and αj = jα, for all α ∈ Q[ξ2d]. Remark 1. If d = 1, then Q[ξ2d, j] ∼= Q(i) and if d 6= 1, denoting w2d = ξ2d + ξ−12d , we have
Q[ξ2d, j] ∼= Q[w2d][ξ2d − ξ−12d , j] ∼= ( (ξ2d − ξ−12d )2,−1
Q[w2d]
) = ( w22d − 4,−1 Q[w2d]
) .
