Two of these groups are isomorphic whenever their Ulm invariants are equal. By a corollary to Schur’s lemma, irreducible representations of these groups are one-dimensional. The torsion elements of one of these groups form a subgroup. In homological algebra, sequences of these groups and homomorphisms between them are the prototypical example of chain complexes. A theorem about these groups is generalized by the structure theorem for finitely generated modules over a principal ideal domain. A subgroup of one of these groups must be a normal subgroup. These groups are equal to their own center. Every finite one of these groups can be expressed as a direct sum of cyclic groups 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. ■END■
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.)
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