Schur  multiplier - 2024 ACF Nationals

Question

For M22 (“M-twenty-two”), the order of this type of corresponding group was first stated by Burgoyne and Fong to be 3, which they then corrected two years later, only for it to be corrected again in 1979. For 10 points each:

[10h] For a given group, Hopf’s formula states that what “multiplier” group is the second homology group with integer coefficients?

ANSWER: Schur (“shoor”) multiplier

[10e] The Schur multiplier of M22 actually has this order. There are this many elements in A4, which corresponds to precisely [emphasize] half the permutations that can act on four objects. You have 10 seconds.

ANSWER: 12 (= 1/2 * 4!)

[10m] The erroneous correction led to an error in the title of the paper that introduced J4, which, like M22, is one of these 26 finite simple groups that do not fit any special pattern.

ANSWER: sporadic groups [or sporadic simple groups or sporadic finite groups]

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Summary


2024 ACF Nationals	2024-04-21	Y	24	11.67	92%	25%	0%

Data


Toronto A	Berkeley A	10	10	20
Yale A	Brown	10	10	20
Chicago C	Cornell B	10	0	10
Harvard	Chicago D	10	0	10
Florida	Claremont Colleges	10	0	10
Indiana	Cornell A	10	10	20
Iowa State	Illinois	10	10	20
Kentucky	McGill	10	0	10
Georgia Tech	Maryland	10	0	10
Michigan	Johns Hopkins	10	0	10
Chicago A	Minnesota A	10	10	20
Northwestern	NYU	10	0	10
North Carolina A	Columbia A	10	0	10
North Carolina B	Berkeley B	10	0	10
Duke	Penn	10	0	10
Texas	Purdue	0	0	0
Stanford	Chicago B	10	0	10
Rutgers	Toronto B	10	0	10
South Carolina	Truman State	10	0	10
Minnesota B	Vanderbilt	10	0	10
Virginia	Columbia B	10	0	10
Arizona State	Yale B	10	10	20
WUSTL A	Waterloo	10	0	10
Ottawa	WUSTL B	0	0	0