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Types of Subgroups in Abstract Algebra

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  • 0:04 What Is a Subgroup?
  • 1:41 Proper & Trivial Subgroups
  • 2:38 The Center of a Group
  • 5:14 Lesson Summary
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Lesson Transcript
Instructor: Michael Gundlach
When looking at groups in abstract algebra, we can often find smaller groups nested within other groups. These smaller groups are called subgroups, and there are some special types of subgroups that are important in abstract algebra.

What is a Subgroup?

Think of the integers. Remember that the set of integers along with the operation of addition form a group. Within this group, there are special integers that we call even integers. Do you remember learning that the sum of two evens is always an even number? This important fact helps show us that even integers form a group all on their own.

The even integers are closed under addition, and we already know that addition is associative, since the integers as a whole form a group. The additive identity is 0, meaning you can add 0 to any number and it keeps its identity. Finally, since a negative even number is still even, the even integers contain inverses. Thus, the even integers under addition form a group.

Whenever a subset of a group is also a group under the same operation, that subset is called a subgroup of the original group. The even integers are a classic example of a subgroup. It turns out that we don't need to test all of the group axioms to determine if a subset is a group. We only need to check if the result of operating two elements from the subset is also in the subset, and if the inverse of any element in the subset is in the subset. Checking these two conditions is called the subgroup test.

The subgroup test is sufficient since we're testing a subset of a known group. Any subset that passes the subset test will contain the identity. Also, the operation in question is known to be associative, since it comes from a known group.

Now that we know what a subgroup is and how to find one, we're going to talk about some basic types of subgroups. These basic types are the proper subgroups, trivial subgroups, and the center.

Proper & Trivial Subgroups

What do you think the largest possible subgroup of a group is? Let's take a quick look back at our definition of a subgroup. We said that a subgroup is a subset of the group that is itself a group. Since every set is a subset of itself, that means that a group is a subgroup of itself. In other words, if we have a group G, G is a subgroup of G. If a subgroup does not contain all of the original group, we call it a proper subgroup. The group of even integers is an example of a proper subgroup.

Now let's determine the smallest possible subgroup. We can make a subgroup by just using {e}, where e is the identity of the original group. Let's check the subgroup test to be sure this is a subgroup. When we operate the identity with itself, we get the identity. The inverse of the identity is the identity. Thus, this small set passed the subgroup test, meaning it is a subgroup. This special, one-element subgroup is called the trivial subgroup.

The Center of a Group

An important subgroup found in every group is the center of the group. The center of a group is the set of all elements that commute with every element in the group. Using symbols, x is in the center of a group G if xg = gx for all gG. Let's check to make sure this set, often represented by Z, is actually a subgroup.

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