#9 ### Group Theory Exercises1. **Subgroup Condition**: Let \( G \) be a group and \( H \) a nonempty subset of \( G \). Show that \( H \leq G \) if \( ab^{-1} \in H \) whenever \( a, b \in H \).2. **Centralizer**: Let \( G \) be a group and \( H \) a subgroup of \( G \). Demonstrate that \( C(H) \leq G \).3. **Intersection of Subgroups**: If \( H_\alpha : \alpha \in A \) are a family of subgroups of the group \( G \), show that \(\bigcap_{\alpha \in A} H_\alpha\) is a subgroup of \( G \).4. **Complex Numbers Subgroup**: Let \( H = \{a + bi \mid a, b \in \mathbb{R}, \, ab \geq 0\} \). Determine whether \( H \) is a subgroup of the complex numbers \( \mathbb{C} \) with addition.5. **Order of Elements**: - Let \( a \) be an element of order \( n \) in a group and let \( k \) be a positive integer. Then \(\langle a^k \rangle = \langle a^{\text{gcd}(n,k)} \rangle \).6. **Order Calculation**: - Let \( a \) be an element of order \( n \) in a group and let \( k \) be a positive integer. Then \(|a^k| = \frac{n}{\text{gcd}(n,k)}\).7. **Cyclic Subgroups**: Every subgroup of a cyclic group is cyclic.8. **Subgroup Lattice**: Determine the subgroup lattice of \( \mathbb{Z}_{36} \).9. **Cyclic Group of Order 3**: Prove that a group of order 3 must be cyclic.

Name: #9 ### Group Theory Exercises1. **Subgroup Condition**: Let \( G \) be
Uploaded: 2024-03-05T11:52:21.000Z
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Description: #9 ### Group Theory Exercises1. **Subgroup Condition**: Let \( G \) be a group and \( H \) a nonempty subset of \( G \). Show that \( H \leq G \) if \( ab^{-1} \in H \) whenever \( a, b \in H \).2. **Centralizer**: Let \( G \) be a group and \( H \)