Group Rings with Solvable Unit Groups of Minimal Derived Length
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Let F be a field of characteristic p > 2 and G a nonabelian nilpotent group containing elements of order p. Write F G for the group ring. The conditions under which the unit group 𝒰(F G) is solvable are known, but only a few results have been proved concerning its derived length. It has been established that if G is torsion, the minimum derived length is ⌈log2(p + 1)⌉, and this minimum occurs if and only if |G′| = p. In the present note, we show that the same holds if G has elements of infinite order.
KeywordsGroup ring Unit group Derived length
Mathematics Subject Classifications (2010)16S34 16U60 20F16
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