-412.4.2 list elements and subgroups oa Z_30

• MHB
• karush
In summary, the subgroups $\langle a^{20}\rangle$ and $\langle a^{10}\rangle$ are the same and contain the elements $\{e,a^{10},a^{20}\}$, where $e$ is the identity element and $a$ has order 30.
karush
Gold Member
MHB
$\tiny{412.4.2}$
(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

Last edited:
karush said:
$\tiny{412.4.2}$
(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)?

1. What is a list element in -412.4.2?

A list element in -412.4.2 is a number or symbol that is part of the sequence -412.4.2, which is a notation used to represent a specific group of numbers.

2. How many list elements are there in -412.4.2?

There are 30 list elements in -412.4.2, as indicated by the subscript number 30 in the notation Z_30.

3. What are subgroups in -412.4.2?

Subgroups in -412.4.2 are smaller groups of numbers that are contained within the larger group represented by -412.4.2. These subgroups have specific mathematical properties and relationships.

4. How are list elements and subgroups related in -412.4.2?

List elements in -412.4.2 are the individual components of the group, while subgroups are subsets of the larger group that share certain characteristics with the main group. Subgroups are made up of list elements.

5. What is the significance of Z_30 in -412.4.2?

Z_30 is a notation used to represent a specific group of numbers, specifically the integers modulo 30. This group has 30 elements and is used in various mathematical applications and calculations.

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