Subgroups

In abstract algebra, subgroups are a concept that enables us to understand and explore the building blocks of groups, one of the fundamental structures in mathematics. Groups themselves provide a way to study symmetries and transformations, which appear in a variety of fields such as physics, chemistry, and computer science. Understanding subgroups is important because they form the basis of many more advanced theories and concepts in algebra, such as factor groups and simple groups.

What is a group?

Before moving on to subgroups, let's quickly recall what a group is. A group is a set G combined with an operation (let's call it *) that satisfies four main properties:

1. Closure: for all elements a, b in G, the result of the operation a * b is also in G.
2. Associativity: for all elements a, b, c in G, the equation (a * b) * c = a * (b * c) is valid.
3. Identity element: There exists an element e in G such that for every element a in G the equation e * a = a * e = a is valid.
4. Inverse element: For every element a in G, there is an element b in G such that a * b = b * a = e, where e is the identity element.

A simple example of a group is the group of integers under addition, where the identity element is 0 and the inverse of any integer a is -a.

Defining a subgroup

A subgroup is a subgroup of a group that is itself a group with the same operation. According to the formal definition, suppose (G, *) is a group. A subgroup H of G is a subgroup of G that forms a group with the operation *. This means that H must satisfy the group properties of closure, associativity, identity, and inverse using the same operation * as G

Notation

If H is a subgroup of G, we write H ≤ G Sometimes, depending on the literature, you may also find H ⊆ G, but with the implicit understanding that H is a subgroup.

Visual example

The larger circle represents the group G, and the smaller circle within it represents the subgroup H

Properties of subgroups

To verify that H is a subgroup of a group G, we use these reduced criteria from the group axioms:

1. Closure: for every pair of elements a, b in H, the product a * b is also in H.
2. Identity: The identity element of G is in H
3. Inverse: For every element a in H, the inverse element a -1 is also in H

Checking for these conditions is often easier than checking for overall group properties.

Example: Subgroup of integers

Consider the group (ℤ, +), which is the set of all integers that are subject to addition. Let's explore some subsets:

• The set of even integers 2ℤ = {..., -4, -2, 0, 2, 4, ...} is a subgroup of ℤ. It satisfies closure (the sum of two even numbers is even), contains the identity (0 is an even integer), and every element has an inverse (if 2a is even, then -2a is also even).
• The set ℤ + = {1, 2, 3, ...} is not a subgroup since it does not contain 0, the identity element, and the reciprocals of positive integers are not positive.

Subgroup testing

An even more sophisticated way to check whether a subset H of G is a subgroup is known as the subgroup test. If H is not empty and for all elements a, b in H, the element a * b -1 is also in H, then H is a subgroup.

Subgroup tests:
H ≠ ∅
2. For all a, b in H, a * b -1 is in H

This test is particularly useful in simplifying proofs, especially in abstract algebraic research.

Viewing subgroup operations

The lines represent the operation * inside the subgroup H, keeping the elements within the subgroup. Here, a * b -1 inside H verifies closure with inverse.

Types of subgroups

There are several special types of subgroups that have unique properties. Let's explore some of these:

Trivial subgroups

The most basic subgroup is called the trivial subgroup. It contains only the identity element of the group, i.e. {e} where e is the identity element of G Every group G has at least one trivial subgroup.

Proper subgroups

A proper subgroup is a subgroup that is not equal to the whole group. If H is a subgroup of G and H ≠ G, then H is a proper subgroup. For example, the group of even integers is a proper subgroup of the integers.

Center of the group

A much more interesting subgroup is the center of a group Z(G). It consists of all elements of G that commute with every other element of G, i.e., Z(G) = {z ∈ G | z * g = g * z for all g ∈ G}. The center of a group is always a subgroup.

Examples and exercises

Let's look at some examples and try to determine which subsets form a subgroup:

Example 1

Consider the set (ℝ*, ⋅) of nonzero real numbers under multiplication. Determine whether the subset ℝ + of positive real numbers is a subgroup.

- Completion: The product of two positive numbers is positive.
- Identity: The identity element is 1, which is positive.
- Reciprocal: The reciprocal of any positive number is also positive.

Thus, ℝ + is a subgroup of ℝ*.

Example 2

Check whether the group of integers under addition, (ℤ, +), contains a subgroup formed by the group of multiples of 3, 3ℤ.

- Completion: The sum of two multiples of 3 is a multiple of 3.
- Identity: the number 0, which is a multiple of 3, is in the group.
- Reciprocal: The reciprocal (under addition) of any multiple of 3 is also a multiple of 3.

Thus, 3ℤ is a subgroup of ℤ.

Exercise 1

Consider the symmetric group S 3, which is the group of all permutations of three elements. List the possible subgroups and verify one of them.

• Subgroups of S 3: the identity {}, the subgroups {(1), (1 2), (2 3), (1 3), (1 2 3), (1 3 2)}, and any group generated by two of these.
• Prove that {(1), (1 2)} is a subgroup.

- Closure: consider each combination of elements (for example, (1 2) * (1 2) = (1)).
- Identity: Identity (1) exists.
- Inverse: Every element is its own inverse.

Thus, {(1), (1 2)} is a subgroup of S 3.

Through these exercises and examples, one can enhance one's understanding of subgroups and their properties within the broader framework of group theory. As you practice identifying and working with subgroups, you will develop a deeper understanding of their role in abstract algebra.

Comments