In the paper 'Groups of units of orders in Q-algebras' by A.L.S. Corner, the following result is proved: A finite group G is realisable as the group of units of an order in a Q-algebra if and only if G is a C2C4C6QDB-group and either (a) G has a direct factor of order 2, or (b) G admits a direct decomposition G=G0×G1×⋯×Gr, where G1,…,Gr are B-blocks and G0 is a C4QD-group which may be embedded as a subdirect product of copies of C4, Q, D in such a way that it contains the diagonal involution −1. The author remarks that the final condition relating to the diagonal involution is not very pretty. He believes that it could be replaced by a more desirable requirement that there exists an element g0 of order 4 in G such that CG(g0) is a 2-group. In the appendix by Federico Menegazzo, the proof of the "more desirable'' requirement is provided.
Appendix to 'Groups of units of orders in Q-algebras'
MENEGAZZO, FEDERICO
2008
Abstract
In the paper 'Groups of units of orders in Q-algebras' by A.L.S. Corner, the following result is proved: A finite group G is realisable as the group of units of an order in a Q-algebra if and only if G is a C2C4C6QDB-group and either (a) G has a direct factor of order 2, or (b) G admits a direct decomposition G=G0×G1×⋯×Gr, where G1,…,Gr are B-blocks and G0 is a C4QD-group which may be embedded as a subdirect product of copies of C4, Q, D in such a way that it contains the diagonal involution −1. The author remarks that the final condition relating to the diagonal involution is not very pretty. He believes that it could be replaced by a more desirable requirement that there exists an element g0 of order 4 in G such that CG(g0) is a 2-group. In the appendix by Federico Menegazzo, the proof of the "more desirable'' requirement is provided.Pubblicazioni consigliate
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