# Basic Group Theory

1. Feb 19, 2005

### rayveldkamp

I lost my notes for the Intro to Group Theory part of my algebra course last year, and need to know a coulple definitions before i go back to uni this year:
ORDER of a group, and
CYCLIC group.
Thanks

Ray Veldkamp

2. Feb 19, 2005

### mathwonk

a cyclic group is one that consists entirely of the powers of a single element, such as 1, x, x^2,x^3,x^4,....,x^n = 1.

this cyclic group has only n elements. thus the order of the group and of the element x is said to be n.

i.e. the order of an element is the smallest power of that elemenmt that equals 1.

this could be infinite. i.e. the integers are an infinite cyclic group, with elements which are not powers but multiples of a single element, namely 1, (for additive groups we say multiples, and for multiplicative groups we say powers).

the order of a group is simply the number of elements in that group.

the order of a group is actually always a multiple of the order of any element in that group.