Let's approach the problem of finding the order of the element $26 + \langle 12 \rangle$ in the factor group $\mathbb{Z}_{60} / \langle 12 \rangle$.

Step 1: Structure of the Factor Group

We are working with the group $\mathbb{Z}_{60}$ (integers modulo 60) and the subgroup $\langle 12 \rangle$, which is the cyclic subgroup generated by 12. The factor group $\mathbb{Z}_{60} / \langle 12 \rangle$ consists of the cosets of $\langle 12 \rangle$ in $\mathbb{Z}_{60}$.

The subgroup $\langle 12 \rangle$ consists of multiples of 12 in $\mathbb{Z}_{60}$, i.e., $\langle 12 \rangle = \{ 0, 12, 24, 36, 48 \}.$

Step 2: The Element and Cosets

The element in question is $26 + \langle 12 \rangle$, which is the coset of 26. Cosets are of the form $a + \langle 12 \rangle$ where $a \in \mathbb{Z}_{60}$. Thus, we are working with the coset: $26 + \langle 12 \rangle = \{ 26, 38, 50, 2, 14 \}.$

Step 3: Finding the Order of the Element

The order of an element $a + \langle 12 \rangle$ in the factor group $\mathbb{Z}_{60} / \langle 12 \rangle$ is the smallest positive integer $n$ such that: $n(26 + \langle 12 \rangle) = \langle 12 \rangle,$ which is equivalent to finding the smallest $n$ such that: $n \times 26 \in \langle 12 \rangle.$ This means we need to find the smallest $n$ such that $n \times 26$ is a multiple of 12 (i.e., $n \times 26 \equiv 0 \pmod{12}$).

Step 4: Determine the Smallest $n$

We need to solve $n \times 26 \equiv 0 \pmod{12}$. First, find the greatest common divisor (gcd) of 26 and 12: $\gcd(26, 12) = 2.$ Since the gcd is 2, the smallest $n$ is $\frac{12}{\gcd(26, 12)} = \frac{12}{2} = 6$.

Conclusion

Thus, the order of the element $26 + \langle 12 \rangle$ in the factor group $\mathbb{Z}_{60} / \langle 12 \rangle$ is 6.

Would you like further details on this process or any related concepts?

Here are 5 related questions:

1. What is the significance of cosets in factor groups?
2. How do you determine the order of an element in a cyclic group?
3. How do greatest common divisors affect the order of elements in modular arithmetic?
4. What is the relationship between subgroups and their respective cosets?
5. How do factor groups simplify the analysis of larger groups?

Tip: When dealing with factor groups, always check the relationship between the generator of the subgroup and the element to determine the smallest multiple leading to the identity element!