#7 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 \( \langle a^k \rangle = \langle a^{\operatorname{gcd}(n, k)} \rangle \).6. Let \( a \) be an element of order \( n \) in a group and let \( k \) be a positive integer. Then \( |a^k| = \frac{n}{\operatorname{gcd}(n,k)} \).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: #7 please 1. Let G be a group and H a nonempty subset of G. Then \( H
Uploaded: 2024-03-20T07:06:41.000Z
Channel: Bartleby
Description: #7 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 o