#5 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 \,|\, 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: #5 1. Let G be a group and H a nonempty subset of G. Then $ H \leq G
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Description: #5 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 subgr

Answered: 5. Let a be an element of order n in a…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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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 \,|\, 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.

Expert Solution

Step 1

To prove : $<a^{k}> = <a^{g c d (n, k)}>$

Let set d = gcd(n,k) and then write k=dr by definition of gcd,

We prove this by showing that each subgroup contains the cyclic generator
of the other side.

( $\subseteq$ )We can say $a^{k} = a^{d r} = (a$ ^d)r

$\Rightarrow$ $a^{k} \in$ $<a^{g c d (n, k)}>$

( $\supseteq$ ) By using gcd as a linear combination.Let d=ns+kr,for some s,r $\in Z$

Then $a^{d} = a^{n s + k r}$ = $(a$ ⁿ)s.(a^k)r $\Rightarrow$ ${(a^{k})}^{r} \in <a^{k}>$

Hence proved.

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