## Abstract

Let k be any field and G a finite group. Given a cohomology class α ∈ H^{2}(G, k*), where G acts trivially on k*, one constructs the twisted group algebra k^{α}G. Unlike the group algebra kG, the twisted group algebra may be a division algebra (e.g. symbol algebras, where G ≅ Z_{n} × Z_{n}). This paper has two main results: First we prove that if D = k^{α}G is a division algebra central over k (equivalently, D has a projective k-basis) then G is nilpotent and G′ , the commutator subgroup of G, is cyclic. Next we show that unless char(k) = 0 and √-1 ∉ k, the division algebra D = k^{α}G is a product of cyclic algebras. Furthermore, if D_{p} is a p-primary factor of D, then D_{p} is a product of cyclic algebras where all but possibly one are symbol algebras. If char(k) = 0 and √-1 ∉ k, the same result holds for D_{p}, p odd. If p = 2 we show that D_{2} is a product of quaternion algebras with (possibly) a crossed product algebra (L/k, β), Gal(L/k) ≅ Z_{2} × Z_{2n}.

Original language | English |
---|---|

Pages (from-to) | 173-198 |

Number of pages | 26 |

Journal | Israel Journal of Mathematics |

Volume | 121 |

DOIs | |

State | Published - 2001 |

Externally published | Yes |