identity element - 2024 ACF Winter at Northwestern

Question

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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Summary


2024 ACF Winter at UC Berkeley	2024-11-16	Y	3	20.00	100%	67%	33%
2024 ACF Winter at Brandeis	2024-11-16	Y	6	20.00	100%	83%	17%
2024 ACF Winter at Clemson	2024-11-16	Y	6	18.33	100%	67%	17%
2024 ACF Winter at Clemson	2024-11-16	Y	2	20.00	100%	100%	0%
2024 ACF Winter at Clemson	2024-11-16	Y	1	20.00	100%	100%	0%
2024 ACF Winter at Clemson	2024-11-16	Y	1	10.00	0%	100%	0%
2024 ACF Winter at Lehigh	2024-11-16	Y	6	13.33	83%	50%	0%
2024 ACF Winter at Northwestern	2024-11-16	Y	9	21.11	89%	89%	33%
2024 ACF Winter at Ohio State	2024-11-16	Y	4	17.50	75%	75%	25%
2024 ACF Winter at Online	2024-11-16	Y	7	14.29	86%	43%	14%
2024 ACF Winter at UBC	2024-11-16	Y	3	20.00	100%	100%	0%
2024 ACF Winter at Oxford	2024-11-16	Y	9	17.78	89%	67%	22%

Data


UIUC D	Indiana B	10	10	10	30
Purdue D	Miami	10	10	0	20
Notre Dame	WashU C	0	0	0	0
UChicago C	Purdue A	10	10	0	20
UChicago A	Purdue B	10	10	10	30
Purdue C	WashU D	10	10	0	20
UIUC A	WashU B	10	10	10	30
UIUC B	Indiana A	10	10	0	20
UIUC C	SIUE A	10	10	0	20