S_3 direct sum A_3/ Z(S_3+ A_3) is isomorphic to which group 

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.

S_3 direct sum A_3 / Z(S_3 + A_3) is isomorphic to which group?

Explore the isomorphism of S_3 direct sum A_3 over the center Z(S_3 ⊕ A_3). This detailed breakdown includes the properties of symmetric and alternating groups, quotient groups, and the role of the group's center. Suitable for university-level abstract algebra students.

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