Let's analyze the expression $S_3 \oplus A_3 / Z(S_3 \oplus A_3)$ to determine which group it is isomorphic to.

Step 1: Understanding the groups involved

• $S_3$ is the symmetric group of degree 3, which consists of all permutations of 3 elements. It has 6 elements: $S_3 = \{ e, (12), (13), (23), (123), (132) \}$

• $A_3$ is the alternating group of degree 3, which is the group of even permutations in $S_3$. It has 3 elements: $A_3 = \{ e, (123), (132) \}$ This is isomorphic to the cyclic group $\mathbb{Z}_3$.

• The direct sum $S_3 \oplus A_3$ consists of ordered pairs where the first element comes from $S_3$ and the second from $A_3$. This group will have $6 \times 3 = 18$ elements.

Step 2: Understanding the center $Z(S_3 \oplus A_3)$

The center $Z(G)$ of a group $G$ consists of all elements that commute with every element of the group.

• For $S_3$, the center is trivial: $Z(S_3) = \{ e \}$, because no non-identity element in $S_3$ commutes with all others.
• For $A_3$, the center is also trivial: $Z(A_3) = \{ e \}$, since $A_3$ is isomorphic to $\mathbb{Z}_3$, a simple group.

Thus, the center of the direct sum $S_3 \oplus A_3$ is also trivial: $Z(S_3 \oplus A_3) = \{(e, e)\}$

Step 3: Simplifying the quotient

The quotient $S_3 \oplus A_3 / Z(S_3 \oplus A_3)$ is essentially the group $S_3 \oplus A_3$ itself, since the center is trivial: $S_3 \oplus A_3 / Z(S_3 \oplus A_3) \cong S_3 \oplus A_3$

Step 4: Isomorphism

We now check what $S_3 \oplus A_3$ is isomorphic to. Since $A_3 \cong \mathbb{Z}_3$, we have: $S_3 \oplus A_3 \cong S_3 \oplus \mathbb{Z}_3$

This is a direct sum of the symmetric group $S_3$ and the cyclic group $\mathbb{Z}_3$. There is no further simplification in terms of common small groups, so the group is isomorphic to $S_3 \oplus \mathbb{Z}_3$.

Conclusion

$S_3 \oplus A_3 / Z(S_3 \oplus A_3) \cong S_3 \oplus \mathbb{Z}_3$

This is the final isomorphic group.

Would you like more details or further clarification?

Related questions:

1. What are the properties of the center of a group?
2. Can the direct product of $S_3$ and $A_3$ be simplified further?
3. How do you find the center of other common groups like $S_n$?
4. What is the structure of $S_n$ for $n > 3$?
5. How does $S_3$ compare to other symmetric groups?

Tip:

In group theory, quotienting by the center of a group gives the inner automorphism group, revealing how elements interact via conjugation.