The notation is just notation. If you like better

or

in lieu of

. They all mean the same thing except that

is reserved for when the groups are abelian (btw

is an analogous index operator for

as that of

for

).

Ok, so for the idea. You need a quick little lemma.

**Lemma:** Let

denote a cyclic (this lemma will show

*the* cyclic) group of order

. Then,

**Proof:** Let

be generated by

and define

by

.

Clearly

is injective since

now we may assume WLOG that

and so the above implies that

. But!

and so

can't be strictly positive and since it's non-negative it follows that

Surjectivity is clear since

for some

and so

and

The homomorphism property follows since

.

The conclusion follows.

Thus, we must only prove that

is cyclic and it will follow from the lemma that it is equal to

To see this we will prove that

generates

. To do this note that

is the least positive number such that

. But, notice that the first coordinate's order is

and so it will only be

when

where

and

is as in

. Similarly, the second slot will only be

when

and so on and so forth. Thus,

is the first occurrence of the multiples of

matching up. In other words

. But, since

and

it follows that

. And since

it follows that

generates it. Thus,

is a cyclic group of order

and thus

by our lemma.