Additive Notation of Abelian Group

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∟Additive Notation of Abelian Group

This section describes the Additive notation of an Abelian Group. The addition sign, +, is used as the operator. Number 0 is used as the identity element.

An Abelian Group can also be expressed in Additive Notation using the addition sign, +, as the operator and number zero, 0, as the identity element:

P + Q                        for F(P,Q)
P + Q = Q + P                for F(P, Q) = F(Q, P)
(P + Q) + S = P + (Q + S)    for F(F(P, Q), S) = F(P, F(Q, S))
P + 0 = P                    for F(P, I) = P
P + Q = 0                    for F(P, Q) = I

The Multiplicative Notation matches well with certain examples of Abelian Groups.

Let's consider the following Abelian Group:

The set of elements is the set of integers.
The binary operation is the numeric addition.
The identity element is the integer 0.

You can verify that all 5 Abelian Group conditions are satisfied. For example:

2 + 3 = 6                   The "closure" condition
2 + 3 = 3 + 2               The "commutativity" condition
(2 + 3) + 4 = 2 + (3 + 4)   The "associativity" condition
3 + 0 = 3                   The "identity" condition
3 + (-3) = 0                The "symmetry" condition