tillsatsgrupp

Tillsatsgrupp is a Swedish term that translates to "additive group" in English. In mathematics, an additive group is a set equipped with a binary operation that satisfies certain properties, specifically those of a group under addition. These properties are closure, associativity, existence of an identity element (the additive identity, usually denoted as 0), and existence of an inverse element for every element (the additive inverse).

The most fundamental example of an additive group is the set of integers (denoted by Z) with the standard operation of addition. This group is infinite and abelian, meaning the order of addition does not matter (a + b = b + a). Other common additive groups include the set of rational numbers (Q), real numbers (R), and complex numbers (C), all under addition.

In abstract algebra, the concept of an additive group is a building block for more complex structures like rings and modules. Rings, for instance, are sets that form an abelian additive group and also have a second binary operation (multiplication) that satisfies distributive laws. Fields are a special type of ring where the non-zero elements form an abelian multiplicative group. Understanding additive groups is therefore crucial for comprehending these advanced algebraic concepts. The notation and properties of additive groups are consistent across various mathematical fields.