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Homework Statement
Determine all integers for which there exists a unique abelian group of order n.
Homework Equations
The Attempt at a Solution
All prime integers?
A unique abelian group of order n is a mathematical structure consisting of a set of n elements and a binary operation that follows the axioms of an abelian group. This means that the operation is commutative, associative, has an identity element, and every element has an inverse. The term "unique" means that there is only one possible group of this order up to isomorphism.
The order of a unique abelian group is simply the number of elements in the group. This can be determined by counting the number of distinct elements or by using the Lagrange's theorem, which states that the order of a subgroup must divide the order of the group.
Yes, a unique abelian group can have non-unique subgroups. This means that there can be multiple subgroups of the same order within the unique abelian group. However, these subgroups will still be isomorphic to each other and to the unique abelian group itself.
Unique abelian groups are a special type of cyclic group. This means that every unique abelian group is also a cyclic group, but the reverse is not necessarily true. Cyclic groups have a single generator, while unique abelian groups can have multiple generators.
Unique abelian groups of order n have applications in many areas of mathematics, including number theory, graph theory, and cryptography. They can also be used to study symmetry and patterns in nature, such as in crystallography and molecular structures. In addition, unique abelian groups have practical applications in coding theory, error-correcting codes, and signal processing.