Subgroup

In group theory, a part of mathematics, a subgroup is a special kind of smaller group inside a bigger group. Imagine you have a big group of friends, and some of those friends form their own smaller group that still follows all the same rules as the big group. That smaller group is called a subgroup.

To be a subgroup, the smaller group must follow the same rules as the bigger group. This means if you take any two members from the smaller group and combine them using the group's rule, the result should also be in the smaller group. Also, if you take any member from the smaller group, its opposite or inverse should also be in the smaller group.

Every group has a very simple subgroup called the trivial subgroup, which includes just the identity element. This is like having a group with only one person who does nothing but stands still. A proper subgroup is a subgroup that is smaller than the whole group, meaning it doesn't include everyone.

Basic properties of subgroups

A subgroup is a smaller group inside a bigger group. The identity element (the special element that does nothing when used in the group operation) in a subgroup is the same as the identity in the bigger group. Also, if an element has an inverse (another element that undoes it) in the subgroup, it has the same inverse in the bigger group.

When you combine two subgroups by taking only the elements they both have, you get another subgroup. However, just putting all elements from two subgroups together does not always make a subgroup. For example, if you take even numbers and multiples of three from the whole number group, their sum can be a number like 5, which is not in either group.

Every element in a group creates a cyclic subgroup by repeating the group operation with itself. This subgroup can be small and finite or infinite, depending on the element.

Cosets and Lagrange's theorem

Main articles: Coset and Lagrange's theorem (group theory)

When we have a smaller group inside a bigger group, we can look at special sets called left cosets. For a smaller group H and an element a in the bigger group G, a left coset is formed by multiplying a with every element in H. These cosets split the bigger group into equal parts.

Lagrange's theorem tells us something important about finite groups. It says that the number of elements in the bigger group divided by the number of elements in the smaller group equals the number of left cosets. This means the size of any smaller group must fit perfectly into the size of the bigger group.

We can also define right cosets in a similar way. If a left coset and a right coset match for every element a, the smaller group is called a normal subgroup.

Example: Subgroups of Z₈

Let's look at a special group called Z₈, which has the numbers 0, 1, 2, 3, 4, 5, 6, and 7. When we add these numbers, we wrap around after reaching 7 (this is called "addition modulo 8").

One subgroup of Z₈ is the set {0, 2, 4, 6}. These are the numbers you get when you multiply 0, 1, 2, 3, 4, 5, 6, and 7 by 2 and then wrap around. This set follows the same addition rules as the whole group, so it is a subgroup.

For any number that divides 8 (like 1, 2, 4, or 8), the multiples of that number in Z₈ form a subgroup. For example:

• Multiples of 1: {0, 1, 2, 3, 4, 5, 6, 7}
• Multiples of 2: {0, 2, 4, 6}
• Multiples of 4: {0, 4}
• Multiples of 8: {0}

These are all the subgroups of Z₈. In fact, for any group like Z_n (where n is a positive whole number), you can find all its subgroups by looking at the multiples of the numbers that divide n. Each of these subgroups will have size n divided by the number you multiplied by.

Example: Subgroups of S₄

The symmetric group S₄ is a group made up of all the ways to rearrange the numbers {1, 2, 3, 4}. A subgroup is a smaller group that follows the same rules as the bigger group.

24 elements

S₄ is a subgroup of itself.

12 elements

The alternating group A₄ is made up of the even permutations in S₄. It is a special subgroup.

8 elements

There are three subgroups with 8 elements, each matching the symmetries of a square.

6 elements

There are four subgroups with 6 elements, each matching the ways to rearrange three numbers while keeping the fourth fixed.

4 elements

There are seven subgroups with 4 elements. Some match a special group called the Klein four-group, and others are made from repeating patterns of four numbers.

3 elements

There are four subgroups with 3 elements, each made from repeating patterns of three numbers.

2 elements

There are nine subgroups with 2 elements. Some are made from swapping two numbers, and others from swapping two pairs of numbers.

1 element

The trivial subgroup is the smallest possible subgroup, containing only one element.

Other examples

Some simple groups can be found inside bigger groups. For example, the even whole numbers make up a group inside all whole numbers because adding or subtracting even numbers still gives an even number.

Also, special parts of number systems or spaces, like certain collections of numbers or points, can act like smaller groups on their own.