1. Showing that $S_3$ is a group with respect to function composition:

The symmetric group $S_3$ is the group of all permutations of three elements, say $\{1, 2, 3\}$. To show that $S_3$ is a group with respect to function composition, we need to check if it satisfies the four group axioms:

• Closure: For any two elements $\sigma$ and $\tau$ in $S_3$, their composition $\sigma \circ \tau$ must also be in $S_3$. Since function composition of permutations is another permutation, closure holds.

• Associativity: Function composition is associative by definition. That is, for any $\sigma, \tau, \rho \in S_3$, we have $(\sigma \circ \tau) \circ \rho = \sigma \circ (\tau \circ \rho)$.

• Identity element: The identity element in $S_3$ is the identity permutation $e = (1)(2)(3)$, which maps each element to itself. For any permutation $\sigma \in S_3$, we have $\sigma \circ e = e \circ \sigma = \sigma$.

• Inverses: For every element $\sigma \in S_3$, there exists an inverse element $\sigma^{-1} \in S_3$ such that $\sigma \circ \sigma^{-1} = \sigma^{-1} \circ \sigma = e$.

Thus, $S_3$ satisfies all the group axioms, and therefore it is a group under function composition.

2. Number of elements in $S_3$ of order 2:

The order of an element $\sigma \in S_3$ is the smallest integer $k$ such that $\sigma^k = e$, where $e$ is the identity permutation.

The elements of $S_3$ are:

1. $e = (1)(2)(3)$ — identity
2. $(12)$ — transposition swapping 1 and 2
3. $(13)$ — transposition swapping 1 and 3
4. $(23)$ — transposition swapping 2 and 3
5. $(123)$ — cyclic permutation: 1 → 2, 2 → 3, 3 → 1
6. $(132)$ — cyclic permutation: 1 → 3, 3 → 2, 2 → 1

Now, we find the order of each element:

• $e$: order 1 (since $e^1 = e$).
• $(12), (13), (23)$: These are transpositions, and each of them squares to the identity: $(12)^2 = e$, $(13)^2 = e$, $(23)^2 = e$. Hence, their order is 2.
• $(123), (132)$: These are 3-cycles, and their order is 3 because $(123)^3 = e$ and $(132)^3 = e$.

Thus, there are 3 elements of order 2: $(12), (13), (23)$.

3. Number of self-invertible elements in $S_3$:

An element $\sigma \in S_3$ is self-invertible if $\sigma = \sigma^{-1}$. This is true if $\sigma^2 = e$. Therefore, the self-invertible elements are those of order 1 or 2.

• The identity $e$ is trivially self-invertible.
• The elements $(12), (13), (23)$, all of order 2, are also self-invertible.

Thus, there are 4 self-invertible elements in $S_3$: $e, (12), (13), (23)$.

Summary:

• $S_3$ is a group under function composition.
• There are 3 elements in $S_3$ of order 2: $(12), (13), (23)$.
• There are 4 self-invertible elements in $S_3$: $e, (12), (13), (23)$.

Would you like further details or clarification?

Here are some related questions:

1. What is the general structure of $S_n$, the symmetric group on $n$ elements?
2. Can you explain why every transposition is of order 2 in any $S_n$?
3. How does the number of elements of each order vary as $n$ increases in $S_n$?
4. What is the significance of self-invertible elements in group theory?
5. Can you describe the concept of conjugacy classes in $S_3$?

Tip: In symmetric groups, transpositions are always of order 2, while cycles of length $k$ have order $k$.