Coset facts for kids

Cosets are a special idea in mathematics that helps us understand how different parts of a group relate to each other. Imagine a group as a collection of things that can be combined, like numbers you can add together, or movements you can make. A coset helps us organize these things into neat, non-overlapping bundles.

To understand cosets, we first need to know what a group and a subgroup are.

What is a Group?

A group in mathematics is a set of items with a way to combine them (like adding or multiplying) that follows specific rules:

• Closure: If you combine any two items in the group, the result is also in the group.
• Identity: There's a special "do nothing" item (like zero for addition or one for multiplication) that doesn't change anything when combined.
• Inverse: For every item, there's another item that "undoes" it, bringing you back to the identity.
• Associativity: When combining three or more items, the way you group them doesn't change the final result.

Think of a clock with numbers 0 to 11. Adding hours on a clock (modulo 12) forms a group. If you add 5 hours and then 8 hours, you end up at 1 o'clock (5 + 8 = 13, and 13 - 12 = 1).

What is a Subgroup?

A subgroup is a smaller group that lives inside a larger group. It uses the same way of combining items as the main group, and it also follows all the group rules.

For example, in our clock group (0 to 11), the set of even numbers {0, 2, 4, 6, 8, 10} forms a subgroup. If you add any two even numbers, you get an even number. Zero is even, and every even number has an "even inverse" (e.g., the inverse of 2 is 10, because 2 + 10 = 12, which is 0 on the clock).

Understanding Cosets

Now, let's talk about cosets. Imagine you have a group G and a subgroup H. A coset is formed by taking every item in the subgroup H and combining it with a specific item (let's call it 'x') from the main group G.

There are two types of cosets:

Left Cosets

A left coset is written as xH. To make it, you take a specific item 'x' from the main group G and combine it on the left with every single item 'h' from the subgroup H.

• So, xH means you get a new set of items: {x combined with h, for every h in H}.

Let's use our clock example.

• Main group G = {0, 1, 2, ..., 11} (addition modulo 12).
• Subgroup H = {0, 2, 4, 6, 8, 10} (even numbers).
• Let's pick an item 'x' from G that is not in H. For example, let x = 1.

The left coset 1H would be:

• 1 + 0 = 1
• 1 + 2 = 3
• 1 + 4 = 5
• 1 + 6 = 7
• 1 + 8 = 9
• 1 + 10 = 11

So, 1H = {1, 3, 5, 7, 9, 11}. This is the set of all odd numbers on the clock.

Notice that the original subgroup H = {0, 2, 4, 6, 8, 10} and the left coset 1H = {1, 3, 5, 7, 9, 11} together make up the entire group G. They are also completely separate (they don't share any items).

Right Cosets

A right coset is written as Hx. To make it, you take every item 'h' from the subgroup H and combine it on the right with a specific item 'x' from the main group G.

• So, Hx means you get a new set of items: {h combined with x, for every h in H}.

Using our clock example again, with H = {0, 2, 4, 6, 8, 10} and x = 1: The right coset H1 would be:

• 0 + 1 = 1
• 2 + 1 = 3
• 4 + 1 = 5
• 6 + 1 = 7
• 8 + 1 = 9
• 10 + 1 = 11

So, H1 = {1, 3, 5, 7, 9, 11}.

In this clock example, 1H and H1 gave us the same set of numbers. This isn't always the case in all groups!

Normal Subgroups

Sometimes, for a specific subgroup H and any item 'x' from the main group G, the left coset xH is exactly the same as the right coset Hx.

• When xH = Hx for every 'x' in G, we say that H is a normal subgroup of G.

Normal subgroups are very important in group theory because they allow us to create new, smaller groups called "quotient groups" or "factor groups." These new groups help mathematicians understand the structure of the original group in a simpler way.

Cosets are like building blocks that help mathematicians break down complex groups into simpler, more manageable pieces. They are used in many areas of mathematics and science, including understanding symmetry in crystals, coding theory, and even in the study of particle physics.

See also

In Spanish: Clase lateral para niños