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The image contains a two-part question related to group theory. Here’s a detailed analysis of each part:

### Question (a):
- **Statement**: Let \( G \) be the permutation group \( S_3 \) (the symmetric group on 3 elements), and \( H \) is the subgroup \( \{ (1), (12) \} \).
- **Task**: Determine if \( H \) is a normal subgroup of \( G \) and provide reasons.

### Solution for Part (a):
1. **Definition of Normal Subgroup**: A subgroup \( H \) of a group \( G \) is normal if for every element \( g \in G \) and \( h \in H \), the element \( g h g^{-1} \in H \). In other words, \( H \) is invariant under conjugation by elements of \( G \).
2. **Elements of \( G = S_3 \)**: \( S_3 \) has 6 elements: 
   \[
   \{ (1), (12), (13), (23), (123), (132) \}
   \]
3. **Subgroup \( H \)**: The subgroup \( H = \{ (1), (12) \} \) has 2 elements, the identity \( (1) \) and the transposition \( (12) \).
4. **Checking Normality**: To verify if \( H \) is normal, compute \( g(12)g^{-1} \) for each \( g \in G \).
   - For some elements of \( G \), like \( (13) \) or \( (23) \), conjugating \( (12) \) will yield elements outside \( H \) (for instance, \( (13)(12)(13)^{-1} = (23) \), which is not in \( H \)).
   - Since conjugation does not keep elements within \( H \), \( H \) is **not a normal subgroup** of \( G \).

### Question (b):
- **Statement**: Let \( G \) be the group \( U(9) \) (the group of units modulo 9) and \( H \) be the normal subgroup \( \{1, 8\} \).
- **Tasks**: 
  1. Choose an element of the factor group \( G/H \) that is not the identity.
  2. Determine its order.

### Solution for Part (b):
1. **Group \( G = U(9) \)**: The group of units modulo 9, \( U(9) \), consists of elements that are coprime to 9 under multiplication modulo 9. Thus, \( G = \{1, 2, 4, 5, 7, 8\} \).
2. **Subgroup \( H = \{1, 8\} \)**: \( H \) is a subgroup of \( G \) and is normal since \( G \) is abelian (all subgroups of an abelian group are normal).
3. **Factor Group \( G/H \)**: The factor group \( G/H \) consists of the cosets of \( H \) in \( G \). These cosets are:
   - \( H = \{1, 8\} \)
   - \( 2H = \{2, 7\} \)
   - \( 4H = \{4, 5\} \)
4. **Choosing a Non-Identity Element**: One possible choice is the coset \( 2H = \{2, 7\} \), which is not the identity coset \( H \).
5. **Order of \( 2H \)**: The order of an element in a factor group is the smallest integer \( n \) such that \( (2H)^n = H \).
   - By calculating, we find that \( (2H)^3 = H \), so the order of \( 2H \) is 3.

Would you like further details on any specific step, or have questions?

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### Related Questions
1. What is the definition of a normal subgroup, and why is it significant in group theory?
2. How do you determine the order of an element in a factor group?
3. Why are all subgroups of an abelian group normal?
4. How does the symmetric group \( S_3 \) differ from other groups in terms of structure and properties?
5. What role does modular arithmetic play in groups like \( U(9) \)?

### Tip
When determining if a subgroup is normal, always check if conjugation by elements in the larger group keeps all subgroup elements within the subgroup.

The image contains a two-part question related to group theory:

(a) Let G be the permutation group S3 and H the subgroup { (1), (12) }. Determine if H is a normal subgroup of G or not. Give reasons.

(b) Let G be the group U(9) and H the normal subgroup {1, 8}. Choose an element of the factor group G/H that is not the identity and determine its order.

Explore whether a subgroup of S3 is normal, and find the order of an element in a factor group of U(9). This group theory problem covers key concepts such as normal subgroups, permutation groups, and the order of elements in factor groups, suitable for college-level mathematics students.

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