MHB -412.4.2 list elements and subgroups oa Z_30

[karush]

$\tiny{412.4.2}$
(a) List the elements of the subgroups $\langle 20\rangle $ and $\langle 10\rangle $ in $\Bbb{Z}_{30}$.
(b) Let $a$ be a group element of order 30.
(c) List the elements of the subgroups $\langle a^{20}\rangle $ and $\langle a^{10}\rangle $.

should be easy ... just never did it

(a) $\Bbb{Z}_{30}=(1,2,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)$

$\langle10\rangle = \{0,10,10^2\}= \{0,10,20\}$

$\langle20\rangle = \{0,20,20^2,20^3,20^4...\}$

$20^2 = 20+20 = 10$ so the elements are $\langle20\rangle = \{0,20,10\}$, same as $\langle10\rangle$ .

kinda maybe

 

[Opalg]

$\tiny{412.4.2}$
(a) List the elements of the subgroups $\langle 20\rangle $ and $\langle 10\rangle $ in $\Bbb{Z}_{30}$.
(b) Let $a$ be a group element of order 30.
(c) List the elements of the subgroups $\langle a^{20}\rangle $ and $\langle a^{10}\rangle $.

should be easy ... just never did it

(a) $\Bbb{Z}_{30}=(1,2,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)$ ? See below.

$\langle10\rangle = \{0,10,10^2\}= \{0,10,20\}$

$\langle20\rangle = \{0,20,20^2,20^3,20^4...\}$

$20^2 = 20+20 = 10$ so the elements are $\langle20\rangle = \{0,20,10\}$, same as $\langle10\rangle$ .

kinda maybe

The group $\Bbb{Z}_{30}$ contains 30 elements. You have only listed 17 of them. Other than that, what you have done so far is correct. So the subgroups $\langle 20\rangle $ and $\langle 10\rangle $ are the same, namely $\{0,20,10\}$. Can you see how that helps with part (c)?