CYCLIC GROUPS 

DEFINITIONS


Definition  A group G is cyclic if there exists g G such that G = {... , g–2, g–1, e, g, g2, ... } - that is, every element of G occurs as a power of g.

The element g is called a generator of G and we write G = g.


Examples

1.  G = {1, i, –1, –i} under multiplication is a cyclic group of order 4.
    G = i = –i.
2.  G = (Z, +) is an infinite cyclic group. G = 1 = –1.
3.  The Klein 4-group, V, is not cyclic: no element generates V.

We can extend this idea. Let G be any group, and let g G.

Then g = {... , g– 2, g–1, e, g, g2, ... } is the smallest subgroup containing g, and is called the cyclic subgroup generated by g.

Definition   Let G be a group, and let g G. The order of g is the smallest positive integer n such that gn = e. If no such n exists, we say that g has infinite order.

We write |g| = n or |g| = . Note that |e| = 1.

Examples

1.  Elements of S4.

    e has order 1
    (1 2) has order 2
    (1 2 3) has order 3
    (1 2 3 4) has order 4
    (1 2)(3 4) has order 2

2.  Elements of Z6.

    As Z6 is a group under addition rather than multiplication, we look at the multiples of an element rather than its powers.

    0: has order 1
    1: 1 + 1 + 1 + 1 + 1 + 1 = 0   order 6
    2: 2 + 2 + 2 = 0                      order 3
    3: 3 + 3 = 0                            order 2
    4: 4 + 4 + 4 = 0                      order 3
    5: 5 + 5 + 5 + 5 + 5 + 5 = 0   order 6.

3.   Elements of {R\{0}, .}.

    If r R has finite order n, then rn = 1.
    Hence |r| = 1 (using | | in the sense of absolute value).
    So r = 1 with order 1, or r = –1 with order 2.
    All other elements have infinite order.