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Every group must consist of a set of elements equipped with an associative binary operation and an element of this type. For 10 points each:
[10e] Elements with what name result in no change when used in a binary operation? This name is also given to matrices with ones on the main diagonal and zeros everywhere else.
ANSWER: identity element [or neutral element; accept identity matrix]
[10m] A theorem named for this man states that for any finite group, the order of any subgroup is an integral divisor of the order of the group. This mathematician also names “multipliers” used for constrained optimization.
ANSWER: Joseph-Louis Lagrange [accept Lagrange’s theorem or Lagrange multipliers]
[10h] For any element x of a group, this is the set of group actions that map x to itself. By a namesake theorem, for any element x of a group, the size of the orbit of x times the size of this set of x is equal to the size of the group.
ANSWER: stabilizer [accept orbit–stabilizer theorem]
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BirminghamSouthampton B1010020
Bristol AManchester100010
Oxford ACambridge A1010020
Cambridge CCambridge B10101030
Durham AWarwick A010010
Cambridge DImperial B1010020
LSE AImperial A100010
LSE BDurham B1010020
Southampton AOxford B10101030
VanderbiltBristol B100010