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Erschienen in:
2018 | OriginalPaper | Buchkapitel
12. ω(G) for Some Classes of Nonabelian Groups
verfasst von : Anthony B. Evans
Erschienen in: Orthogonal Latin Squares Based on Groups
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Abstract
Almost all attempts to improve the lower bounds for ω(G) have been for abelian groups, in particular direct products of elementary abelian groups. The only classes of nonabelian groups for which attempts have been made to improve the lower bound for ω(G) are the dihedral groups and some of the linear groups of even characteristic. We will present these improvements in this chapter. We will derive improved lower bounds for ω(D2n), D2n the dihedral group of order 2n: these improvements will be obtained from improved lower bounds for \(\omega (\mathbb {Z}_n)\) and improved lower bounds for ω(D2n) when n is a power of 2. While proofs of admissibility show that ω(G) ≥ 1 when G is nonsolvable, very little work has been done on improving this bound. We will show that \(\omega (G)\ge \min \{\omega (\mathbb {Z}_{q-1}),\omega (\mathbb {Z}_{q+1})\}\), when G = GL(2, q) or SL(2, q), q even, q≠2. To establish this bound, we will use partitions of the element sets of the groups. For higher dimensions we will establish lower bounds for ω(GL(n, q)), q even, q≠2, using even-odd decompositions and partitions of the sets of odd-order elements of the groups.