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- Jun 22, 2012

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"Notation: For each \(\displaystyle n \in \mathbb{Z}^+ \), let \(\displaystyle Z_n \) be the cyclic group of order n

__. " (my emphasis)__

**(written multiplicatively)**But this notation is surely a bit counter-intuitive since \(\displaystyle Z_n \) is an additive group.

Indeed, Gallian defines/explains \(\displaystyle Z_n \) in EXAMPLE 2, page 74, as follows:

**EXAMPLE 2**The set \(\displaystyle Z_n = \{ 0,1, 2, ... \ ... ,n-1 \} \) is a cyclic group under addition modulo n.

Surely the Gallian notation and explanation of the group is clearer.

The D&F definition/notation leads to the need for constant vigilance as in the example in Chapter 3, page 74, illustrating quotient groups where \(\displaystyle Z_n \) is defined as \(\displaystyle <x> \) with elements \(\displaystyle x^a \) - but of course the elements are actually of the form \(\displaystyle x + x + ... \ ... + x \) (a terms). Then we read statements like (see attachment - top of page 75)

"The multiplication in \(\displaystyle Z_n \) is just \(\displaystyle x^a x^b = x^{a+b} \)"

Why use "multiplication" to describe an operation that is actually addition?

Surely this is not intuitive - nor is it pedagogically helpful.

What do forum members think?

Can someone explain the likely motivation for D&F adopting this notation? What are the advantages of such a notation?

Peter