Group Theory Basics: Order & Cyclic Groups

[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

 

[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.

 

[M98Ranger]

Sure, I'd be happy to help refresh your memory on the basics of group theory.

First, let's define the order of a group. The order of a group is simply the number of elements in the group. For example, if we have a group of integers under addition, the order would be infinite since there are an infinite number of integers. However, if we have a group of integers modulo 5 under multiplication, the order would be 4 since there are only 4 elements in the group (1, 2, 3, and 4).

Next, let's discuss cyclic groups. A cyclic group is a group where all the elements can be generated by a single element, called the generator, through repeated application of the group operation. This means that if we take the generator and perform the group operation with itself multiple times, we will eventually get all the elements in the group. An example of a cyclic group is the group of integers under addition. The generator in this case would be 1, and by adding 1 to itself multiple times, we can generate all the integers in the group. Another example is the group of rotations in a square, where the generator would be a 90 degree rotation.

I hope this helps refresh your memory on these concepts. Good luck with your studies!