Symmetric Groups

students, imagine a lock with several numbered keys 🔑. A symmetric group is the math language for every possible way to rearrange those keys. This lesson explains what symmetric groups are, why they matter, and how they fit into the bigger study of permutation groups. By the end, you should be able to identify the symmetric group on a set, describe its elements using cycle notation, and understand how composition and inverses work.

What is a symmetric group?

A permutation is a rearrangement of the elements of a set. If a set has $n$ elements, then the symmetric group on that set is the set of all permutations of those elements. It is usually written as $S_n$ when the set has $n$ objects.

For example, if the set is $\{1,2,3\}$, then $S_3$ contains every possible rearrangement of these three numbers. There are $3! = 6$ of them. The symbol $n!$ means the product $n(n-1)(n-2)\cdots 2\cdot 1$.

So for $S_3$, the six permutations are:

the identity permutation, which keeps everything fixed
the three swaps of two elements
the two 3-cycles, which rotate all three elements

A useful way to think about $S_n$ is this: it is the collection of all ways to shuffle $n$ distinct items. If you line up $n$ books on a shelf, every possible ordering of those books is one element of $S_n$ 📚.

Why is it called a group?

A symmetric group is not just a set of permutations. It is a group because it satisfies the group axioms:

Closure: If you combine two permutations, the result is still a permutation.
Associativity: Composition of functions is associative.
Identity element: The identity permutation does nothing and is in the group.
Inverses: Every permutation can be undone by another permutation.

The group operation in $S_n$ is composition of functions. If $\sigma$ and $\tau$ are permutations, then $\sigma\tau$ means “do $\tau$ first, then do $\sigma$.” This order matters because function composition is not usually commutative.

For example, in $S_3$, let $\sigma = (12)$ and $\tau = (23)$. Then:

$\sigma\tau$ means apply $(23)$ first, then $(12)$.
$\tau\sigma$ means apply $(12)$ first, then $(23)$.

These can give different results, which shows that $S_n$ is generally non-abelian when $n \ge 3$.

Understanding elements of $S_n$ with cycle notation

To write permutations clearly, mathematicians use cycle notation. A cycle shows how elements move in a repeating loop.

For example, the cycle $(123)$ means:

$1 \mapsto 2$
$2 \mapsto 3$
$3 \mapsto 1$

This is a 3-cycle. It can also be written as a function on the set $\{1,2,3\}$.

A transposition is a cycle of length $2$, such as $(12)$, which swaps $1$ and $2$ and leaves everything else fixed.

The identity permutation is sometimes written as $e$ or as a cycle with no movement. In $S_3$, the identity keeps $1$, $2$, and $3$ in place.

Cycle notation makes permutations easier to read because it focuses on what changes. For instance, in $S_5$, the permutation $(135)(24)$ means:

$1 \mapsto 3$, $3 \mapsto 5$, $5 \mapsto 1$
$2 \mapsto 4$, $4 \mapsto 2$

These two cycles are disjoint because they move different elements. Disjoint cycles can often be understood separately, which makes them very useful in calculations.

Products of permutations

A key skill in symmetric groups is multiplying permutations. Because the group operation is composition, you must be careful about the order.

Suppose in $S_3$ we have $\sigma = (12)$ and $\tau = (123)$. To compute $\sigma\tau$, start with each element and apply $\tau$ first, then $\sigma$.

Let’s track each number:

$1 \xrightarrow{\tau} 2 \xrightarrow{\sigma} 1$, so $1$ is fixed
$2 \xrightarrow{\tau} 3 \xrightarrow{\sigma} 3$, so $2 \mapsto 3$
$3 \xrightarrow{\tau} 1 \xrightarrow{\sigma} 2$, so $3 \mapsto 2$

So $\sigma\tau = (23)$.

Now reverse the order and compute $\tau\sigma$:

$1 \xrightarrow{\sigma} 2 \xrightarrow{\tau} 3$, so $1 \mapsto 3$
$2 \xrightarrow{\sigma} 1 \xrightarrow{\tau} 2$, so $2$ is fixed
$3 \xrightarrow{\sigma} 3 \xrightarrow{\tau} 1$, so $3 \mapsto 1$

Thus $\tau\sigma = (13)$.

Since $\sigma\tau \ne \tau\sigma$, the order of multiplication matters. This is one of the main reasons symmetric groups are so important in abstract algebra: they give a clear example of a group that is usually not abelian.

Inverses in symmetric groups

Every permutation has an inverse, meaning another permutation that reverses the rearrangement.

If $\sigma$ sends $a$ to $b$, then $\sigma^{-1}$ sends $b$ back to $a$.

For cycles, the inverse is easy to find: just reverse the direction of the cycle. For example:

$$(123)^{-1} = (132)$$
$$(12)^{-1} = (12)$$

A transposition is its own inverse because swapping two elements twice brings everything back to the original position.

More generally, if a permutation is written as a product of disjoint cycles, the inverse is found by inverting each cycle separately. For example,

$\big((135)(24)\big)^{-1} = (135)^{-1}(24)^{-1} = (153)(24).$

This works because the inverse of a composition reverses the order of the factors.

A real-world analogy is undoing a series of moves in a puzzle game 🎮. If you rotate tiles and then swap two pieces, the inverse process must reverse both actions in the opposite order.

The size and structure of $S_n$

The symmetric group on $n$ objects has exactly $n!$ elements. This is because:

there are $n$ choices for the first position,
then $n-1$ choices for the second,
then $n-2$ choices for the third,
and so on until all positions are filled.

So the number of permutations is:

$|S_n| = n!$

Some small examples are:

$|S_1| = 1$
$|S_2| = 2$
$|S_3| = 6$
$|S_4| = 24$

As $n$ grows, the number of permutations grows very quickly. This makes symmetric groups rich and complex.

Another important fact is that $S_n$ contains many different kinds of subgroups. For example, the set of even permutations forms a subgroup called the alternating group $A_n$. Also, the subgroup generated by a single permutation can reveal patterns in repeated movement.

Understanding $S_n$ is important because it is one of the most concrete examples of a group. Many abstract ideas in group theory can be tested and visualized using permutations.

How symmetric groups fit into permutation groups

The topic of permutation groups is broader than symmetric groups. A permutation group is any group made of permutations of a set, with composition as the operation.

In that broader setting, $S_n$ is the most complete example: it contains every permutation of an $n$-element set. So every subgroup of $S_n$ is also a permutation group.

This means symmetric groups act like a “home base” for the study of permutations. If you want to understand how permutations behave, $S_n$ is the natural place to start. From there, you can study:

subgroups of $S_n$
cycle structure
order of permutations
parity of permutations
actions on sets

For example, the symmetries of a square can be described by a subgroup of $S_4$. Each symmetry rearranges the four corners of the square, so it can be viewed as a permutation. This shows how symmetric groups connect abstract algebra to geometry and symmetry in the real world 🟦.

Conclusion

students, symmetric groups are the groups of all permutations of a set. They are written as $S_n$ and contain exactly $n!$ elements. Their operation is composition, their elements can be written in cycle notation, and every element has an inverse. Symmetric groups are central to permutation groups because they provide the largest and most familiar example of a permutation group. Learning $S_n$ gives you tools for understanding structure, symmetry, and non-commutative behavior in abstract algebra.

Study Notes

A permutation is a rearrangement of a set.
The symmetric group on $n$ elements is written as $S_n$.
$S_n$ contains all permutations of an $n$-element set.
The group operation in $S_n$ is composition of functions.
Composition is read right to left: in $\sigma\tau$, do $\tau$ first.
$S_n$ has $n!$ elements, so $|S_n| = n!$.
Cycle notation shows how elements move in a permutation.
A transposition is a cycle of length $2$.
The inverse of a cycle is found by reversing the cycle direction.
A permutation and its inverse undo each other.
For $n \ge 3$, $S_n$ is generally non-abelian.
Symmetric groups are the main example of permutation groups and help connect abstract algebra to real symmetry situations.