Multiplicative Notation of Abelian Group

EC Cryptography Tutorials - Herong's Tutorial Examples

∟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 * 1 = 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