Subgroups for $F=(\mathbb Z_{12},+)$

Asked Jan 12 '21 at 17:01

Active Jan 12 '21 at 17:11

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I'm working with the following.

Find all subgroups for $F=(\mathbb Z_{12},+)$
Which subgroups are cyclic groups?

$1,5,7,11$ generate $F$. $$\left \langle [2] \right \rangle = \left \{[0], [2], [4], [6], [8], [10] \right \}\\ \left \langle [3] \right \rangle = \left \{[0], [3], [6], [9] \right \}\\ etc.$$

How do I know which are cyclic groups?

Bonus questions:

If it would be subgroups for $F=(Z_{12},\cdot)$, would the subgroups be...? $$\left \langle [2] \right \rangle = \left \{[0], [2], [4], [8] \right \}\\ \left \langle [3] \right \rangle = \left \{[0], [3], [9] \right \}$$

edited Jan 12 '21 at 17:11

Shaun

44,997

asked Jan 12 '21 at 17:01

user9060784

1

a subgroup of a cyclic group is cyclic; and $[0]$ is not an element of $(\mathbb Z_{12},\cdot)$ – J. W. Tanner Jan 12 '21 at 17:07
All subgroups of cyclic groups are cyclic. Reference: https://math.stackexchange.com/q/295564/104041 – Shaun Jan 12 '21 at 17:08
1

To start with, look for the subsets of $F$ closed under addition modulo $12$. – Jan 12 '21 at 17:09

Subgroups for $F=(\mathbb Z_{12},+)$

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