DEFINITIONS
Definition A group G is cyclic if there exists g G such that G = {... , g2, g1, 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
We can extend this idea. Let G be any group, and let g Then Definition Let G be a group, and let g We write |g| = n or |g| =
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.
G.
g
= {... , g 2, g1, e, g, g2, ... } is the smallest subgroup containing g, and is called the cyclic subgroup generated by 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.
. Note that |e| = 1.
1. Elements of S4. e has order 1 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. 3. Elements of {R\{0}, .}. If r
Examples
(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
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.
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.