(This post is part of the algebra notes series.)

These notes are based on the book Contemporary Abstract Algebra 7th ed.

Terminology and Notation

Order of a Group

The number of elements of a group (finite or infinite) is called its order. |G| denotes the order of G.

For example, U(10) = {1,3,7,9} is of order 4.

Order of an Element

The order of an element g in a group G is the smallest positive integer n such that gⁿ=e. If no such integer exists, we say g has infinite order.

Examples

Subgroup

If a subset H of a group G is itself a group under the operation of G, we say that H is a subgroup of G.

Subgroup notation

H <= G means H is a subgroup of G. If we want to specify that H is a subgroup of G but not equal to G, we write H<G to indicate a proper subgroup.

The subgroup {e} is the trivial subgroup of G.

Subgroup Tests

To prove if a subset of G is a subgroup of G, we don't have to verify all the group axioms (closure, associativity, identity, inverses). We can use special subgroup tests.

One-Step subgroup Test

Let G be a group and H a nonempty subset of G. If ab^-1 is in H whenever a,b are in H, then H is a subgroup of G.

Examples using the one-step subgroup test

When proving a group H is a subgroup of G, the very first thing you do is show H is nonempty. After that, you suppose you have elements a and b in H and attempt to show that ab^-1 is in H. When you are supposing that a and b are in H, the very first thought you should have is "What does it mean for these elements to be in H? What do they look like? What properties to they have?". These properties are usually spelled out in the definition of the group itself. In the above example, the property of elements of H was that x²=e. Almost immediately after supposing a and b were in H, we spelled out what exactly that meant in terms of what those elements looked like. Without taking the time to evaluate what these elements look like, we have no clear direction to take the proof.

The next thought you should have is "How do I know when I've arrived? When am I finished? What is the end goal?". For the one-step subgroup test, the overall goal is to show that ab^-1 is in H. However, similar to when you let a and b be in H, you should ask yourself "What does it mean for ab^-1 to be in H? What does it look like? What properties does it have?". In the above example, since elements of H had the property that x²=e, we concluded that if ab^-1 were to actually be an element of H, it would need to have the property that (ab^-1)²=e, and verifying that would finish the proof. By taking the time to figure out what the end of the proof would look like, we created a road map for ourselves to help us solve the problem.

Trying to prove something while not having a sense of direction for the proof is a recipe for frustration.

Here is another example which ends up being a bit similar.

If you're not certain why we get to say that xy^-1 is an element of G, it's because we're guaranteed closure by the fact that G is a group.

Next time

In the next article of the series, we'll discuss the two-step subgroup test, the finite subgroup test, and provide many more examples of subgroups.

(This post is part of the algebra notes series.)