Index in Abstract Algebra

students, imagine a club where some students are in a special subgroup, like the chess team inside the whole school. A big question is: how many different “copies” of that subgroup can we make by shifting it around inside a larger group? That idea is called the index. It is one of the key ways cosets help us count structure inside groups 🔎

What you will learn

• What the index of a subgroup means
• How to use cosets to count the index
• How index connects to the size of a group and the size of a subgroup
• Why index matters in Lagrange’s Theorem
• How to use examples to find index in practice

What is Index?

In group theory, suppose $G$ is a group and $H$ is a subgroup of $G$. The index of $H$ in $G$ is the number of distinct left cosets of $H$ in $G$. It is written as $[G:H]$.

So, if we list all the different sets of the form $aH$ for $a \in G$, the number of distinct sets is the index. These cosets partition the whole group, meaning every element of $G$ belongs to exactly one left coset of $H$.

Think of it like organizing all students in a school into equal-sized teams based on a club. If each team has the same number of students as the club members, then the number of teams tells you how many “copies” of the club fit into the whole school. That count is the index 🎯

If $G$ is finite, then the index is a whole number. If $G$ is infinite, the index may still be a number, but in some cases it can be infinite too.

A simple example

Let $G = \mathbb{Z}$, the group of integers under addition, and let $H = 3\mathbb{Z}$, the subgroup of multiples of $3$.

The left cosets are:

• $0 + 3\mathbb{Z}$, which is the set of integers divisible by $3$
• $1 + 3\mathbb{Z}$, which is the set of integers congruent to $1$ mod $3$
• $2 + 3\mathbb{Z}$, which is the set of integers congruent to $2$ mod $3$

These are the only distinct cosets, so the index is $[\mathbb{Z}:3\mathbb{Z}] = 3$.

This example shows the core idea: the index counts the distinct cosets, not the size of the subgroup itself.

Index and Cosets

To understand index well, students, you need to understand cosets. A left coset of a subgroup $H$ in a group $G$ is a set of the form $aH = \{ah : h \in H\}$ for some $a \in G$.

A right coset is a set of the form $Ha = \{ha : h \in H\}$.

In general, left and right cosets may be different, especially when the group is not abelian. However, the index of a subgroup is usually defined using left cosets, and in finite groups the number of left cosets equals the number of right cosets.

Each left coset has exactly the same number of elements as $H$, because the map $H \to aH$ given by $h \mapsto ah$ is a bijection. That fact is very important. It means every coset is a shifted copy of the subgroup.

Why this matters

If $G$ is finite, then the cosets of $H$ divide up the whole group into equal-sized pieces. If the subgroup has size $|H|$ and there are $[G:H]$ cosets, then the total size of the group is

$$|G| = [G:H] \cdot |H|$$

This formula is the heart of Lagrange’s Theorem.

Example in a finite group

Let $G = \mathbb{Z}_6 = \{0,1,2,3,4,5\}$ under addition mod $6$, and let $H = \{0,3\}$.

The left cosets are:

• $0 + H = \{0,3\}$
• $1 + H = \{1,4\}$
• $2 + H = \{2,5\}$

After that, the pattern repeats because $3 + H = \{3,0\} = H$, so there are only three distinct cosets.

Thus $[\mathbb{Z}_6 : H] = 3$.

Check the counting:

• $|G| = 6$
• $|H| = 2$
• $[G:H] = 3$

And indeed $6 = 3 \cdot 2$.

Index and Lagrange’s Theorem

Lagrange’s Theorem says that if $G$ is a finite group and $H$ is a subgroup of $G$, then $|H|$ divides $|G|$.

Why? Because the group is split into disjoint cosets of equal size, and the number of those cosets is the index $[G:H]$. So we get

$$|G| = [G:H]|H|$$

This shows that the index is not just a counting idea; it is the number that makes the size equation work.

Consequences of Lagrange’s Theorem

If $|G|$ is finite, then possible subgroup sizes are limited by the divisors of $|G|$. For example, if $|G| = 12$, then a subgroup can only have order $1,2,3,4,6,$ or $12.

The index also has a relationship to the subgroup size:

• If $|G| = 12$ and $|H| = 3$, then $[G:H] = 4$
• If $|G| = 12$ and $|H| = 4$, then $[G:H] = 3$

So knowing the order of a subgroup can help you find its index, and knowing the index can help you find the subgroup order.

Example with symmetry

Consider the group of symmetries of a square, which has $8$ elements. A subgroup might be the rotations, which has $4$ elements. Then the index is

$$[G:H] = \frac{8}{4} = 2$$

That means there are exactly two cosets of the rotation subgroup inside the whole symmetry group. This tells us the rotations form a subgroup that sits halfway inside the full symmetry structure.

How to Find Index in Practice

Here is a useful procedure students can use:

1. Identify the group $G$ and the subgroup $H$
2. Determine the cosets of $H$ in $G$
3. Count the distinct cosets
4. That count is the index $[G:H]$

If the group is finite, you can often use the formula

$$[G:H] = \frac{|G|}{|H|}$$

as long as you already know $H$ is a subgroup of $G$.

Example: subgroup in $\mathbb{Z}_{12}$

Let $G = \mathbb{Z}_{12}$ and let $H = \{0,4,8\}$.

The cosets are:

• $0 + H = \{0,4,8\}$
• $1 + H = \{1,5,9\}$
• $2 + H = \{2,6,10\}$
• $3 + H = \{3,7,11\}$

Then the pattern repeats, so there are $4$ distinct cosets. Therefore

$$[\mathbb{Z}_{12} : H] = 4$$

This also matches the size formula:

$$\frac{12}{3} = 4$$

Infinite example

Let $G = \mathbb{Z}$ and $H = 5\mathbb{Z}$.

There are five distinct cosets:

• $0 + 5\mathbb{Z}$
• $1 + 5\mathbb{Z}$
• $2 + 5\mathbb{Z}$
• $3 + 5\mathbb{Z}$
• $4 + 5\mathbb{Z}$

So the index is

$$[\mathbb{Z} : 5\mathbb{Z}] = 5$$

This shows index can be finite even when the group itself is infinite.

Why Index Matters

Index is useful because it measures how a subgroup sits inside a larger group. A subgroup of small index is “close” to the whole group, while a subgroup of large index appears in many separate cosets.

It also helps in proving deeper results. For example, if a subgroup has index $2$, then it is automatically normal. That fact comes from the way left and right cosets line up when there are only two cosets.

Index also appears in many parts of algebra beyond this lesson, including permutation groups, quotient groups, and counting arguments in finite group theory.

For students, the most important idea is this: index is the number of equal-sized coset pieces that partition the group 🧩

Conclusion

students, the index of a subgroup is the number of distinct left cosets of that subgroup in the whole group. It connects directly to cosets because it is defined by counting them. It connects to Lagrange’s Theorem because, in a finite group, the group order equals the subgroup order times the index. That gives the formula

$$|G| = [G:H] |H|$$

Understanding index helps you count, compare subgroup sizes, and predict what subgroup structures are possible. It is one of the most important ideas linking cosets, subgroup order, and the size of the whole group.

Study Notes

• The index of a subgroup $H$ in a group $G$ is written $[G:H]$.
• $[G:H]$ equals the number of distinct left cosets of $H$ in $G$.
• Every left coset has the same number of elements as $H$.
• For a finite group, $$|G| = [G:H] \cdot |H|$$
• This formula is the key idea behind Lagrange’s Theorem.
• In finite groups, the order of a subgroup must divide the order of the group.
• Index can be found by counting cosets or by using $[G:H] = \frac{|G|}{|H|}$ when the group is finite.
• A subgroup of index $2$ is always normal.
• Index helps show how cosets partition a group into equal-sized pieces.