Question

Two of these groups are isomorphic whenever their Ulm invariants are equal. By a corollary to Schur’s lemma, irreducible representations (-5[1])of these groups are one-dimensional. (10[1])The torsion elements of one of these groups form a subgroup. In (10[1])homological algebra, sequences of these groups (10[1])and homomorphisms between them are the prototypical example of chain complexes. A theorem about these groups is generalized by the structure theorem (-5[1])for finitely (10[1])generated (10[1])modules over a principal ideal domain. (10[1]-5[1])A subgroup of one of these groups must be a normal subgroup. (-5[1])These (10[1])groups (10[1])are equal to their own center. (10[3])Every finite one of these groups can be expressed as a direct sum (-5[1])of cyclic groups (10[1])of prime power order. One example of these groups is the integers under addition. For 10 points, name these groups whose binary operation is commutative. (10[1])■END■ (10[8]0[2])

ANSWER: abelian (“uh-BEEL-ee-in”) groups [accept commutative group until “commutative” is read; until “representation” is read, prompt on countable group or p-groups by asking “what other type of group is being referred to?”; until “finite” is read, prompt on finite groups by asking “what other type of group is being referred to?”] (The theorem in the fifth sentence is the fundamental theorem of finite abelian groups, which is stated in the eighth sentence.)
<Other Science>
= Average correct buzz position

Summary

2023 ACF Nationals04/22/2023Y2295%0%23%99.19

Buzzes

PlayerTeamOpponentBuzz PositionValue
Kevin YeUC Berkeley BTexas A19-5
Swapnil GargUC Berkeley AIowa State A2410
Alex AkridgeIndiana AYale B3610
Henry CafaroChicago CGeorgia Tech A4210
Richard NiuCornell BNorthwestern A64-5
Vivek SasseChicago BCornell A6610
Max ChemtovMcGill ARutgers B6710
Raymond JiangGeorgia Tech BBrown A73-5
Tim MorrisonStanford ANYU A7310
Gio MazzeoVirginia ANorth Carolina A85-5
Maxwell YeMinnesota ARutgers A8610
Forrest WeintraubColumbia BToronto A8710
Simon GorbatyDuke AIllinois A9310
Tanis NielsenClaremont AMinnesota B9310
Geoffrey WuColumbia AYale A9310
Adam FineChicago AVanderbilt A106-5
Isaac MammelMaryland APenn A10910
Raul PassementTexas AUC Berkeley B13410
Aidan FeinVanderbilt AChicago A13510
Arjun NageswaranHarvard AHouston A13510
Vincent DuNorth Carolina AVirginia A13510
Anson BernsBrown AGeorgia Tech B13510
Andrew SalijNorthwestern ACornell B13510
Alex SchmidtPenn State AMIT A13510
Matthew LehmannWUSTL APurdue A13510
Todd MaslykMichigan AFlorida B1350
Jeya IyaduraiFlorida BMichigan A1350
Vedul PalavajjhalaWUSTL BOhio State A13510