#6 please 1. Let \( G \) be a group and \( H \) a nonempty subset of \( G \). Then \( H \leq G \) if \( ab^{-1} \in H \) whenever \( a, b \in H \).2. Let \( G \) be a group and \( H \) a subgroup of \( G \). Show that \( C(H) \leq G \).3. 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. 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. Let \( a \) be an element of order \( n \) in a group and let \( k \) be a positive integer. Then \( \langle a^k \rangle = \langle a^{\gcd(n,k)} \rangle \).6. Let \( a \) be an element of order \( n \) in a group and let \( k \) be a positive integer. Then \( |a^k| = \frac{n}{\gcd(n,k)} \).7. Every subgroup of a cyclic group is cyclic.8. Determine the subgroup lattice of \( \mathbb{Z}_{36} \).9. Prove that a group of order 3 must be cyclic.

Name: #6 please 1. Let \( G \) be a group and \( H \) a nonempty subset of \
Uploaded: 2024-03-20T07:07:37.000Z
Channel: Bartleby
Description: #6 please 1. Let \( G \) be a group and \( H \) a nonempty subset of \( G \). Then \( H \leq G \) if \( ab^{-1} \in H \) whenever \( a, b \in H \).2. Let \( G \) be a group and \( H \) a subgroup of \( G \). Show that \( C(H) \leq G \).3. If \( H_{\a