An additive group is a group, and any group can be written as an additive group, so the adjective "additive" does not describe a class of groups, but rather the notation used to write the group operation. This is a stumbling block for many beginners, who are not used to this level of abstraction. The term group can refer to several concepts: Look up Group in Wiktionary, the free dictionary In music, a group is another term for band or other musical ensemble. ...

This is best illustrated by way of an example. There is only one group with exactly three elements and, like all groups, many different notations can be used to describe the same mathematical object. Written as an additive group, this group is usually called Z ₃, the integers modulo three. The elements of the group are written 0, 1, 2 and the group operation is written +. It is writing the operation + that makes the notation "additive". In this notation, we write that in 2 + 2 = 1 (that is, 1 is the remainder when 2 + 2 is divided by 3). The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ... The word modulo is the Latin ablative of modulus. ... The term element can refer to: Chemical element — material that consists of atoms with the same number of protons in the nucleus (see also Periodic table). ...

This exact same group can also be written as a multiplicative group. In that case, the group operation is written * or is "understood" through concatenation and the elements are usually written as "e" (the identity element), x, and x². In this case the exact same equation as the one at the end of the previous paragraph is written x² * x² = x, or even (using concatination) x² x² = x. In mathematics, multiplicative group in group theory may mean any group G written in multiplicative notation (rather than additive notation for an abelian group) for its binary operation or in particular the multiplicative group of a field F, namely F{0} under multiplication, written F* or Fx. ... In formal language theory (and therefore in programming languages), concatenation is the operation of joining two character strings end to end. ... In philosophy, it is important to distinguish between two senses of identity, qualitative identity and numerical identity. ...

Understanding that the mathematical objects and the operations are the same, independent of the notation, is the biggest hurdle that beginners studying abstract algebra must leap. Abstract algebra is the field of mathematics concerned with the study of algebraic structures such as groups, rings and fields. ...