Exploration of cyclic groups

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This survey article explores cyclic groups from a number of viewpoints, including their occurrence as quotients of the group of integers, modular arithmetic, their role as subgroups in any group.

Modular arithmetic and the role of cyclic groups

Arithmetic modulo n is addition of integers in the ordinary fashion, except that we set n to be equal to zero. Intuitively, every time we reach n, we wrap back to zero. Here are some examples.

Time: Clocks, calendars

If you go round once, you're back to where you started.
Seven days a week, twelve hours a clock, twenty-four hours a day

There are twelve hours on the wall-clock. The twelve hours are denoted by the positive integers 1,2,3,4,5,6,7,8,9,10,11,12. These denote hour positions.

Suppose we start the hour hand pointing at 12. Moving the hour hand forward by 2 hours and then moving it forward by 3 hours moves the hour hand forward by 5 hours, so that it points at 5. More generally, we expect that if we move the hour hand forward by a hours and then move it forward by b hours, it has moved forward by (a + b) hours, so it points at (a + b). However, if a + b is bigger than 12, then we wrap around the clock once, so we effectively reach a + b - 12. Thus, moving the hour hand forward by 7 and then moving it forward by 8 makes it point to 7 + 8 - 12 = 3 hours.

We can describe this by an arithmetic of addition, where the set of numbers is 0,1,2,3,4,5,6,7,8,9,10,11. To add two numbers here, we add them as integers. If the ordinary sum is less than 12, then that's the sum we want. If the ordinary sum is at least 12, then we subtract 1 to get the desired sum.

This is the arithmetic of addition mod 12.

Under this addition, the numbers 0,1,2,3,4,5,6,7,8,9,10,11 form an Abelian group. The associativity of addition is clear from the definition: the order in which one parenthesizes the operations of moving the hour hand forward, doesn't matter. Commutativity is also clear: what order you do the two operations doesn't matter. The identity operation is As for inverses, if you move the hour hand forward by a hours, then moving it forward by 12 - a hours does the trick of reversing the operation. This Abelian group is termed the cyclic group of order 12, denoted \mathbb{Z}/12\mathbb{Z} or \mathbb{Z}_{12} or C_12.

A similar phenomenon can be seen in days of the week. Going forward by 2 days and then going forward by 3 days, has the effect of going forward by 2 +3 = 5 days of the week. But going forward by 3 days and then 4 days gets one back to the same day of the week. Thus, the addition here is modulo 7, and the arithmetic of addition again gives an Abelian group.

Some other examples:

  1. There are 60 minutes in an hour. The operations of moving the minute hand forward, then, form a cyclic group of order 60.
  2. There are 24 hours in a day. If we're working with a 24-hour clock, the operations of moving the hour hand forward, form a cyclic group of order 24.
  3. There are 12 notes in an octave. The operations of transposing the notes, i.e., moving them up on the scale, gives a cyclic group of order 12.

Why have wrap-arounds and modular arithmetic?

Wrap-around and modular arithmetic is good for a number of reasons. In some cases, it is forced upon us by nature. For instance, the sun rises once a day, so we need a way to wrap around and get back to where we started once a day. Next, we divide the day into convenient intervals (like hours). The number of intervals that we use (which is, in this case, 24) tells us what size of cyclic group we get.

This is natural for time intervals because of the natural time periodicity of the motions of heavenly bodies: for instance, the earth rotates once around its axis every day, and revolves once around the sun every year. However, the choice of how many intervals to divide it into is arbitrary and based on our convenience rather than for any intrinsic reason.

In the case of music, both the natural periodicity of music and the choice of dividing into 12 parts are natural in some sense. Further information: cyclic groups in music

Modular arithmetic

Modular arithmetic is the study of cyclic groups: groups that we obtain from the integers by wrapping some integer n to zero. n = 7 corresponds to days of the week, n = 60 corresponds to minutes in an hour, n = 24 corresponds to hours in a day, and so on. Modular arithmetic seeks to develop a general theory to study cyclic groups modulo n for different n.

The general description for integers modulo n

The cyclic group of order n (n is termed the order of the group, and also called the modulus or the period of addition) is described as follows:

  • Its elements are the integers 0,1,2,3,\dots,n-1
  • The sum of two elements a and b is their usual integer sum a + b if a + b < n, and is a + b - n if a + b \ge n
  • The identity element for addition is 0
  • The inverse of 0 is 0, and the inverse of a is 12 - a for other a

A somewhat better description

Though the above is the easiest way to describe the cyclic group, the following description is somewhat better conceptually. Here, we think of the cyclic group as the set of all integers, except that we declare two integers to be equal if and only if they differ by a multiple of n. This is saying that you could move the hour hand forward by 1000 hours, but moving it forward by 1000 hours is equivalent to moving it forward by 4 hours (because 1000 - 4 is a multiple of 12).

In the language of sets and equivalence relations, we're taking the set of all integers, and introducing an equivalence relation on the set, where two integers are equivalent if their difference is a multiple of n. In other words, two integers are equivalent if they leave the same remainder modulo n. This equivalence relation is termed being congruent modulo n. In symbols, we write:

a \equiv b \mod n

if a and b are congruent mod n.

The relation of being congruent mod n is preserved by addition, i.e., if:

a \equiv b \mod n, \qquad c \equiv d \mod n

Then:

a + c \equiv b + d \mod n

Thus, the cyclic group mod n that we have is a group whose elements are the equivalence classes. We're adding equivalence classes.

What we effectively did in the previous description was to simply replace each equivalence class by the unique element in that class among 0,1,2,\dots,n-1: the remainder there.