heltalsgruppe

Heltalsgruppe is a term used in number theory and abstract algebra to describe a specific type of mathematical structure. It refers to a set of numbers, usually integers, that are equipped with an operation, typically addition or multiplication, which satisfies certain group axioms.

A group, in general, is a set together with a binary operation that combines any two elements to form a third element, satisfying four fundamental properties: closure, associativity, the existence of an identity element, and the existence of an inverse element for each element in the set.

When these properties are applied to the set of integers with the operation of addition, the resulting structure is known as the additive group of integers, often denoted as (Z, +). Here, closure means that the sum of any two integers is also an integer. Associativity holds because (a + b) + c = a + (b + c) for any integers a, b, and c. The identity element is 0, as a + 0 = a for any integer a. The inverse of any integer 'a' is '-a', since a + (-a) = 0.

Similarly, the set of integers with the operation of multiplication forms a structure that is *not* a group. While closure, associativity, and an identity element (1) exist, not every integer has a multiplicative inverse within the integers. For example, 2 does not have an integer inverse such that 2 * x = 1.

Therefore, "heltalsgruppe" most commonly refers to the additive group of integers. This concept is foundational in abstract algebra and is a building block for understanding more complex algebraic structures. It is a cyclic group, meaning all its elements can be generated by a single element.