Prüfer

Prüfer is a German surname most commonly associated in mathematics with Heinz Prüfer, a German mathematician who contributed to combinatorics and algebra. The name is linked to two standard mathematical constructions: the Prüfer code and the Prüfer p-group.

Prüfer code: In graph theory, the Prüfer sequence provides a one-to-one correspondence between labeled trees on n vertices and sequences of length n−2 formed from the vertex labels 1 through n. Each labeled tree corresponds to a unique sequence, and each such sequence determines a tree. This bijection implies that the number of labeled trees on n vertices is n^(n−2) and underpins methods for counting, encoding, and generating trees algorithmically.

Prüfer p-group: In abelian group theory, the Prüfer p-group (also called the quasi-cyclic p-group) is the unique p-divisible group that is the union of cyclic groups of order p^k under inclusion. It is denoted by Z(p^∞) or C_{p^∞}. The Prüfer p-group serves as a standard example of an injective object in the category of abelian p-groups and appears in the study of p-divisible groups and module theory.

Beyond these constructions, the Prüfer surname appears in various mathematical contexts through related works and terminology inspired by Heinz Prüfer. The term Prüfer thus denotes influential concepts in discrete mathematics and algebra, reflecting the impact of his contributions on these fields.