Groups with a nontrivial nonideal kernel

Abstract

A b s t r ac t . We classify finite groups G, such that the group algebra, QG (over the field of rational numbers Q), is the direct product of the group algebra Q[G/N ] of a proper factor group G/N , and some division rings.2010 Mathematics Subject Classification. 20C15.

Classification of affine symmetry groups of orbit polytopes

“…The next lemma can often be used to show that a certain group is not the affine symmetry group of a rational orbit polytope. As a consequence of the classification of the finite groups G with NKer Q (G) = 1 [14,Theorem D], it turns out that most of these groups satisfy the assumptions of the next lemma. 7.1.…”

“…When G is not the affine symmetry group of an orbit polytope with lattice points as vertices, then NKer Q (G) = 1. The list of such groups consists of the groups in the above list, and the following groups: (a) G = Q 8 × C 4 × (C 2 ) r × A, (b) G = Q 8 × Q 8 × (C 2 ) r × A, where in each case A is abelian of odd order, and the multiplicative order of 2 modulo |A| is odd [14,Theorem D]. However, these groups can be realized as symmetry groups of integer orbit polytopes.…”

“…As an example, consider the group G = Q 8 × C 7 . Then it is known that QG is a direct product of division rings [14,26]. This is essentially due to the fact that Q(ε), where ε is a primitive 7-th root of unity, is not a splitting field of the quaternions over Q. Equivalently, −1 is not a sum of two squares in Q(ε).…”

“…This is essentially due to the fact that Q(ε), where ε is a primitive 7-th root of unity, is not a splitting field of the quaternions over Q. Equivalently, −1 is not a sum of two squares in Q(ε). More generally, for a field k, we have that kG is a direct product of division rings if and only if −1 is not a sum of two squares in k(ε) [14,Theorem 4.2]. When kG is not a direct product of division rings, then kG contains a simple left ideal which is not a twosided ideal and on which G acts faithfully.…”