Introduction to Abstract Algebra: Part 2 - Cyclic Groups and Subgroups

jackeown (48)in #mathematics • 7 years ago (edited)

Introduction to Abstract Algebra: Part 2

Last year I took the classes Abstract Algebra I and II from the great Joseph Gallian who literally wrote the book on abstract algebra. I loved the classes and learned a lot. This is why I decided to write this series of posts.

Part One - Introduction to Groups
Part Two - Cyclic Groups and Subgroups
Part Three - Review and the Symmetric Group

Review of Cyclic Groups!

In Part One I mentioned cyclic groups. They look like {0,1,2,3,4,...,n-1} with the operation of addition modulo n. They are referred to as "Zn" or sometimes "Cn".

This is a group and having addition be modular is necessary for it to be finite and still a group. If you didn't have the modulus, then in Z5 you could do 4+3=7 and 7 isn't in {0,1,2,3,4} (which breaks closure). However...with the modulus, 4+3=2 which IS in {0,1,2,3,4} and therefore Closure isn't broken!

A cyclic group is called called cyclic for this very nature, but this technically isn't the only type of cyclic group. A cyclic group is either some Zn...OR...It could be the set of ALL THE INTEGERS! (Z)

What really makes a group "cyclic" is the ability to be "generated" from a single element!
...let's explore what this means...

Group "Order" and "Generators"

The order of a group is the "size" of that group...(the number of elements in it as a set).
However, elements within a group also have something we call an "order."
The order of an element, a, is the smallest natural number, b, for which a^b = e.

(recall that "e" is the identity of the group and that "^" is repeated application of our group operation!)

The order of an element, G, is written |G| and the order of an element, a, is written |a|.

If every element of a group can be found by taking different powers of some element, then that group is called cyclic and the element is called a generator.
It should make sense that a group is cyclic if and only if it contains an element of the group which has order equal to that of the whole group.

Cyclic Subgroups

If a subset of elements from a group also satisfy all of the group properties, then that subset of elements can be called a "subgroup" of the original group.

If a group has some element, a, then <a> refers to the subgroup formed by taking all integer powers of a.
This is called the cyclic subgroup generated by a.
(BTW a^(-3) means the inverse of a combined with itself and the group operation 3 times and for FINITE cyclic groups it will always be able to be written as a positive exponent instead.)

Examples!

Z15 is the set {0,1,2,3,4,5,6,7,8,9,10,11,12,13,14}.
1 is a generator since every element, a, can be written as 1^a.

(Remember that "*" is our operator [even though it's basically addition in this case] and "^" means repeated application of "*" so 1^5 = 11111 = 5)

In fact, every number relatively prime to 15 will generate this group!
Let's try...<4> = {0,4,8,12,1,5,9,13,2,6,10,14,3,7,11}.
That's the whole group!
Therefore not only is 1 a generator, but 4 is also! (as is everything relatively prime to 15...)

There's also a unique subgroup for every divisor of the order of the group! (For finite abelian [commutative] groups only)
Let's try...
<3> = {0,3,6,9,12} <-- order 5
<5> = {0,5,10} <-- order 3

(Note that 3^5=15=0 and we've started wrapping around forever so we'll only get these five elements)