Multiplicative Notation of Abelian Group

This section describes the multiplicative notation of an Abelian Group. The multiplication sign, *, is used as the operator. Number 1 is used as the identity element.

An Abelian Group can be expressed in Multiplicative Notation using the multiplication sign, *, as the operator and number one, 1, 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 * I = P                    for F(P, I) = P
P * Q = 1                    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 all rational numbers without 0.
• The binary operation is the numeric multiplication.
• The identity element is the number 1.

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

```2 * 3 = 6                   The "closure" condition
1/6 * 1/6 = 1/3             The "closure" condition
2 * 3 = 3 * 2               The "associativity" condition
(2 * 3) * 4 = 2 * (3 * 4)   The "commutativity condition
3 * 1 = 3                   The "identity" condition
3 * 1/3 = 1                 The "symmetry" condition
```

Last update: 2019.