To find the order of the element $5 + \langle 4 \rangle$ in the factor group $\mathbb{Z}_{12} / \langle 4 \rangle$, follow these steps:

1. Identify the structure of the factor group $\mathbb{Z}_{12} / \langle 4 \rangle$:

   • $\mathbb{Z}_{12}$ consists of elements $\{0, 1, 2, \ldots, 11\}$.
   • $\langle 4 \rangle$ is the subgroup generated by 4 in $\mathbb{Z}_{12}$. The elements of $\langle 4 \rangle$ are $\{0, 4, 8\}$ because: $4 \times 0 \equiv 0 \pmod{12}, \quad 4 \times 1 \equiv 4 \pmod{12}, \quad 4 \times 2 \equiv 8 \pmod{12}, \quad 4 \times 3 \equiv 12 \equiv 0 \pmod{12}$
   • So, $\mathbb{Z}_{12} / \langle 4 \rangle$ consists of the cosets $\{ \langle 4 \rangle, 1 + \langle 4 \rangle, 2 + \langle 4 \rangle, 3 + \langle 4 \rangle \}$.

2. Find the coset $5 + \langle 4 \rangle$:

   • $5 + \langle 4 \rangle = \{ 5, 9, 1 \}$ because: $5 \equiv 5 \pmod{12}, \quad 5 + 4 \equiv 9 \pmod{12}, \quad 5 + 8 \equiv 13 \equiv 1 \pmod{12}$

3. Determine the order of the element $5 + \langle 4 \rangle$:

   • The order is the smallest positive integer $k$ such that $k(5 + \langle 4 \rangle) = \langle 4 \rangle$.
   • Compute: $2(5 + \langle 4 \rangle) = 10 + \langle 4 \rangle$
     • $10 + \langle 4 \rangle = \{ 10, 2, 6 \}$, which is not $\langle 4 \rangle$. $3(5 + \langle 4 \rangle) = 15 + \langle 4 \rangle \equiv 3 + \langle 4 \rangle$
     • $3 + \langle 4 \rangle = \{ 3, 7, 11 \}$, which is not $\langle 4 \rangle$. $4(5 + \langle 4 \rangle) = 20 + \langle 4 \rangle \equiv 8 + \langle 4 \rangle$
     • $8 + \langle 4 \rangle = \langle 4 \rangle$.

   Therefore, the order of $5 + \langle 4 \rangle$ is 4.

Summary: The order of the element $5 + \langle 4 \rangle$ in $\mathbb{Z}_{12} / \langle 4 \rangle$ is 4.

Would you like any further details or explanations?

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Related Questions:

1. How do you determine the order of an element in an arbitrary factor group?
2. What is the significance of the factor group $\mathbb{Z}_{12} / \langle 4 \rangle$?
3. How can you generalize the process to find orders of elements in different cyclic groups?
4. What is the relationship between the order of an element and the subgroup it generates?
5. Can you explain how to identify all cosets of a subgroup in a given group?

Tip: When dealing with factor groups, remember that the order of an element is the same as the smallest multiple that brings it back to the identity coset.