Sylow theorems - 2024 PLAYTIME

Question

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]

<JC, Other Science - Math>

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2025 PLAYTIME @ CWRU	12/10/2024	Y	5	16.00	100%	40%	20%
2025 PLAYTIME @ Claremont	12/10/2024	Y	5	24.00	100%	80%	60%
2025 PLAYTIME @ Florida	12/10/2024	Y	3	16.67	100%	33%	33%
2024 PLAYTIME @ Harvard	04/26/2025	Y	4	22.50	100%	75%	50%
2024 PLAYTIME @ JHU	10/26/2024	Y	8	16.25	88%	63%	13%
2025 PLAYTIME @ Northwest	12/10/2024	Y	4	12.50	75%	25%	25%
2025 PLAYTIME Online	04/26/2025	Y	9	20.00	100%	67%	33%
2025 PLAYTIME @ Sheffield	12/10/2024	Y	6	18.33	100%	83%	0%
2024 PLAYTIME @ UBC	10/05/2024	Y	8	21.25	100%	88%	25%
2024 PLAYTIME @ Virginia Tech	07/11/2025	Y	3	20.00	100%	67%	33%
2024 PLAYTIME @ WashU	10/12/2024	Y	3	16.67	100%	33%	33%

Data


Michigan A	Michigan D	0	10	0	10
Michigan B	Pittsburgh	10	10	10	30
Ohio State A	Oberlin	0	10	10	20
Ohio State D	Ohio State B	0	10	0	10
Michigan C	Ohio State C	0	10	0	10
Caltech	UCSD	10	10	10	30
Claremont A	UCLA B	0	10	10	20
Art of Premature Signaling	Claremont B	10	10	10	30
UCLA A	girl 15 whale i mean they	10	10	10	30
P.C.‚ Daniel‚ Michel‚ and Jason B	UCSB	0	10	0	10
Edwin Laundrorat's Long-Lost Man Unclothès Rat Degrosses	Keeping Up With the Cardassians	0	10	0	10
Project2025BC: Make Assyria Great Again	Billy's Bastion	10	10	0	20
Qin Shi Huang's Immortality Poison	Too much from you lately	0	10	10	20
RIT	Belmont	10	10	0	20
Brown	Brandeis	10	10	10	30
Harvard	Lexington	0	10	10	20
MIT	BU	0	10	10	20
George Washington	George Mason B	0	10	0	10
George Mason C	Haverford A	0	0	0	0
Haverford B	Catholic University of America	0	10	10	20
NYU A	Johns Hopkins A	0	10	10	20
NYU B	Johns Hopkins B	0	10	10	20
Maryland A	Georgetown	10	10	10	30
George Mason A	NYU C	0	10	0	10
Rutgers	Maryland B	0	10	10	20
Carleton	SD Mines	0	0	0	0
Washington Connections	Alberta	0	10	0	10
Arizona State University A	UW A	10	10	10	30
we need new names	Egypt Airlines Haters	0	10	0	10
Four Generalists in Search of a Power	2 Italians, 2 Poles, All Noles	0	10	0	10
MSJ	Gruh	0	10	0	10
Free Agents 1	Illinois Orange	10	10	10	30
Auburn 2026	Mahjonng Masters	10	10	10	30
Mike Dukakis Tank Battalion	mongeese	10	10	10	30
Notre Dame + Epic Games	Monsieur Hulot's Academic Team	0	10	10	20
Trip-Hop Quizzer Tricky	Quatumvirate	0	10	10	20
evil women enjoyers	Goober Gang	0	10	10	20
UW B	Arizona State University B	0	10	0	10
Warwick C	Sheffield A	0	10	10	20
Cambridge	Warwick A	0	10	10	20
Edinburgh	Manchester	0	10	10	20
Lancaster	Imperial A	0	10	0	10
Imperial B	Sheffield C	0	10	10	20
Sheffield B	Warwick B	0	10	10	20
Simon Fraser	Mac A	10	10	10	30
McGill B	McGill A	0	10	10	20
Toronto A	Guelph	0	10	10	20
Toronto B	Toronto D	0	10	0	10
Toronto C	Ottawa A	10	10	10	30
Waterloo A	UBC	0	10	10	20
Toronto E	Waterloo B	0	10	10	20
Waterloo C	Toronto Metropolitan	0	10	10	20
Even in Blacksburg There I Am	4 quizbowl pros go Virginia on North Carolina	10	10	10	30
Adrianne Lenker 4 Nobel Lit Prize 2K65	Master and Margaritaville	0	10	10	20
Ashurbanipal’s Drag Race	Somehow Paula Deen Returned	0	10	0	10
Iowa State	Chicago B	10	10	10	30
Missouri	WashU Bulbasaur	0	10	0	10
WashU Charmander	Chicago A	0	10	0	10