What follows is a rather rough draft. It's a basic introduction into something of what cyclic groups are. Perhaps I'll add more some day!
I'm going to have to get a bit technical here, but bear with me - it's not so bad really!
Group tables (such as the ones below) are good for an introduction to group theory, but they quickly become unwieldy when you get down the road with a few more ideas. You can still use them, they just get rather large (often infinite...) and they can obscure clarity of vision for generalizations.
The usual alternative is a system of notation called "presentations", and cyclic groups are useful objects for introducing the style.
A group G may be written as
G = < X | R >
where X is a set of generators, and R is a set of relators. A generators can
be any element of the group except for the identity (the identity as I'm sure
you'll know is often referred to as "e" in group tables. It's usually
referred to as "1" in groups defined by presentations). Relators are equations
containing some of the generators as their "variables" (or more correctly
"indeterminates") and tell you when two sequences of generators produce the
same group element.
OK, time for an example.
The cyclic group C_2 ("C subscript 2") can be presented as follows: C_2 = < x | x² = 1 >. This means that it has a single generator, "x", and when you multiply (that is, "do the group operation with") x and x you get back to the identity. The group table for this would be
| 1 x -+---- 1| 1 x x| x 1For C_3 we have C_3 = < x | x³ = 1 >
| 1 x x² --+--------- 1 | 1 x x² x | x x² 1 x²| x² 1 x
The generalization should now be clear!
Cyclic groups can be visualized a bit like clock faces. The hours in the 12-hour clock form C_12 (and the 24 hour clock form C_24), but this time the group operation would more logically be called "addition". For example 5 (from 12 ie 5 o'clock) + 8 (hours round past 12 sort of) = 1 (from 12 ie 1 o'clock). 12 is the identity here - we ought to take to writing it as "0"! Draw some clock faces with different numbers of hours and you get different cyclic groups. When I was at school we called this "modulo arithmetic" or even "clock arithmetic".
Going back to C_3 (group operation is called "multiplication" again...) we could write x² = x^-1, that is "x squared = x inverse (1/x)". It shouldn't take you long to see why this makes sense, because x² x = x³ = 1. Armed with this idea of group inverses as being the same as multiplicative inverses, the whole thing about groups and fields begins to take more flesh onto the bones. The additive inverse of eg 4 is of course -4, and immediately we could renumber the hours on the standard clock face: we identify 12 with 0 and write 12->0, and we also have: 1->1, 2->2, 3->3, 4->4, 5->5 and 11-> -1, 10-> -2, 9-> -3, 8-> -4, 7-> -5. Since 6 = -6 we may as well retain 6->6.
Make sense? If so stretch out the clock from between the 6 and the -5. Make the face a bigger circle, adding the appropriate numbers as you go. Up goes the order of your cyclic group. Keep stretching (in your imagination!) until the circle becomes a straight line going off infinitely in both directions. You now have what could be called the "cyclic group of infinite order", but is most often referred to as "the free group of order 1". You'll see that it is simply the integers, and it's canonical (standard) presentation is < x | >. No relators because you never get back to the identity - that's why it's called a "free group", because there are no relators ("order 1" counts the number of generators). By the way, I don't want to talk about whether or not the two lines of the "clock face" "join up at infinity"...!