Hw9 Problem 1

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### Problem 1

List the elements of $\mathbb{Z_2} \times \mathbb{Z_4}$, along with each of their orders. Is the group cyclic?

**Solution**

$\langle (0,0) \rangle = \{(0,0)\}$ order of 1

$\langle (0,1) \rangle = \{(0,0), (0,1), (0,2), (0,3)\}$ order of 4

$\langle (0,2) \rangle = \{(0,0), (0,2)\}$ order of 2

$\langle (0,3) \rangle = \{(0,0), (0,1), (0,2), (0,3)\}$ order of 4

$\langle (1,0) \rangle = \{(0,0), (1,0)\}$ order of 2

$\langle (1,2) \rangle = \{(0,0), (1,2)\}$ order of 2

$\langle (1,3) \rangle = \{(0,0), (0,2), (1,1), (1,3)\}$ order of 4

Group is not cyclic because 2 and 4 are not relatively prime.