Packet 8: Tossup 7

A finite extension of the rationals has a property of this name if and only if it is contained in a cyclotomic field. The Mordell-Weil theorem (-5[1])applies to structures like elliptic curves that have a property of this name (15[2])when considered as varieties. (15[1])For a group to be solvable, each quotient group in its subnormal series (-5[1])must have a (-5[1])property of this name. (15[1])If a group has a property of this name, its torsion elements form a subgroup, and it equals its own (*) center. (10[3])By a fundamental (10[2])theorem, (10[1])finite groups with that property of this name (10[1]-5[1])are (10[1])the direct sum of cyclic (10[1])groups of prime power order. (10[1])If a group has a property of this name, its Cayley table is symmetric and any conjugation is the identity; (10[1])such groups include (10[1])Z/nZ (“Z-mod-n-Z”). For 10 points, a group whose binary operation is commutative (10[3])has what (10[1])property named for a (10[1])Norwegian (10[1])mathematician? (10[1])■END■ (10[3]0[1])

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