Answer to Question #313206 in Abstract Algebra for Jyo

Question

In (𝑍, +) , let 𝐻 = 𝑠𝑒𝑡 𝑜𝑓 𝑎𝑙𝑙 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑒𝑠 𝑜𝑓 3 and

𝐾 = 𝑠𝑒𝑡 𝑜𝑓 𝑎𝑙𝑙 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑒𝑠 𝑜𝑓 5.

Show that H and K are subgroups of Z . Also describe 𝐻 ∩ 𝐾

Answer 1

We shall show that H and K are closed under + and every element in H and K have their respective inverses in H and K

Let $h,h°∈H$ . Then $h=3k$ and $h°=3q$ , where $k,q∈ℤ$

$h+h°=3k+3q=3(k+q)$

let $k+q=m∈ℤ$

Hence, H is closed under +

Let $h°$ be the inverse of $h∈H$

$=>h+h°=0$ (identity element)

$=>3k+h°=0$

$=>h°=-3k=3(-k)$

Let $-k=n∈ℤ$

Thus, $h°∈H$

Hence, $(H,+)$ is a subgroup of $(ℤ,+)$

Let $k,k°∈H$ . Then $k=5r$ and $k°=5q$ , where $r,q∈ℤ$

$k+k°=5r+5q=5(r+q)$

let $r+q=m∈ℤ$

Hence, K is closed under +

Let $k°$ be the inverse of $k∈H$

$=>k+k°=0$ (identity element)

$=>5r+k°=0$

$=>k°=-5r=5(-r)$

Let $-r=n∈ℤ$

Thus, $k°∈H$

Hence, $(K,+)$ is a subgroup of $(ℤ,+)$

H∩K={z∈ℤ: z=15m, m∈ℤ}