Feit and Thompson showed every finite group whose order has this property is solvable. For 10 points each:
[10m] Name this property possessed by permutations whose sign is negative. Functions with this property satisfy [read slowly] f of minus x equals minus f of x.
ANSWER: oddness
[10h] A group is solvable if it has a sequence described by the sub-form of this property whose quotients are abelian. Quotient groups are well defined for subgroups described by this word, which are invariant under conjugation.
ANSWER: normal subgroups [accept subnormal sequence]
[10e] All subgroups are normal for abelian groups, whose binary operations have this property. Matrix multiplication lacks this property, which a binary operation has if a times b always equals b times a.
ANSWER: commutativity [or commutative]
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