The integers are one of these structures under addition, but not under multiplication.
HNN extensions can be used to prove that every constructible one of these structures can be defined using a finite presentation. A homomorphism from one of these structures to a general linear one of them on a vector space is one of their namesake representations. If one of these structures possesses a single element in its generating set it is said to be (*) cyclic. These structures are equivalent to monoids which possess an inverse. The integers are one of these structures under addition, but not under multiplication. For 10 points, name these algebraic structures characterized by a set and an operation which together possess closure, associativity, an inverse element, and an identity element, whose commutative varieties are named for Abel.
HNN extensions can be used to prove that every constructible one of these structures can be defined using a finite presentation.
groups