The second of these theorems states that all members of a namesake class of maximal subgroups are conjugate to each other. For 10 points each:
[10h] Name these three theorems about subgroups of a finite group of prescribed order, first proved by a Norwegian mathematician.
ANSWER: Sylow theorems
[10e] The Sylow theorems concern Sylow p-subgroups of a group, which are maximal subgroups whose order is a power of one of these numbers. Every integer greater than 1 has a unique factorization in terms of these numbers by the fundamental theorem of arithmetic.
ANSWER: prime numbers
[10m] The third Sylow theorem states that the number of Sylow p-subgroups n has this relationship to the order of the group m. A prime element has this relationship to A or B whenever it has this relationship to A times B, which is denoted by a vertical bar between p and AB.
ANSWER: divides [accept answers indicating that p is a divisor or a factor of AB]
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