Question

All the quotient groups are cyclic and therefore isomorphic to \(\mathbb{Z}_{n}\) for some \(n\). In each case, find this \(n\). $$ \mathbb{Z}_{15} /\langle 6\rangle $$

Short answer

The quotient group is isomorphic to \( \mathbb{Z}_5 \).

Step-by-step solution

Identify the Groups and Subgroups Involved

We are given the quotient group \( \mathbb{Z}_{15} / \langle 6 \rangle \). Here, \( \mathbb{Z}_{15} \) is the group of integers modulo 15, and \( \langle 6 \rangle \) is the subgroup generated by 6 in \( \mathbb{Z}_{15} \). This subgroup contains all multiples of 6 modulo 15.

Determine the Order of the Subgroup

The order of the subgroup \( \langle 6 \rangle \) is determined by the smallest positive integer \( m \) for which \( 6m \equiv 0 \pmod{15} \). We need to determine the order of the element 6 in \( \mathbb{Z}_{15} \).

Calculate the Greatest Common Divisor (GCD)

To find when \( 6m \equiv 0 \pmod{15} \), we need the greatest common divisor gcd(6, 15). This value will affect how often we cycle through multiples of 6 until reaching a multiple of 15 (remainder 0). Calculate gcd(6, 15) = 3.

Compute the Order of the Quotient Group

The number of distinct cosets in \( \mathbb{Z}_{15} / \langle 6 \rangle \) is given by the index of \( \langle 6 \rangle \) in \( \mathbb{Z}_{15} \). This is the size of \( \mathbb{Z}_{15} \) divided by the size of the subgroup \( \langle 6 \rangle \), which is \( \frac{15}{3} = 5 \). Thus, the quotient group is isomorphic to \( \mathbb{Z}_5 \).

Conclusion

The quotient group \( \mathbb{Z}_{15} / \langle 6 \rangle \) is isomorphic to \( \mathbb{Z}_5 \). The integer \( n \) for which the quotient group is \( \mathbb{Z}_n \) is 5.

Key concepts

Cyclic Group

A cyclic group is a group that can be generated by a single element. This means every element of the group can be expressed as a power (or multiple) of this generator. Cyclic groups are important because they are simple and have a predictable structure. They can be either infinite or finite.
For finite cyclic groups, like \( \mathbb{Z}_n \), each element is some integer multiple of the generator and can be written in modular arithmetic.

• Generator: The generator of a cyclic group is a key element from which the whole group is derived.
• Order of a cyclic group: The order is the number of distinct elements in the group.

The beauty of cyclic groups is that they are straightforward to understand once you comprehend the generator and order.

Group Theory

Group theory is the mathematical study of symmetry and structure within a set with a binary operation. A group is a set equipped with an operation satisfying four main properties: closure, associativity, identity, and invertibility.
Groups are used in many fields of mathematics to solve equations, model natural phenomena, and understand abstract concepts.

• Closure: For all elements a, b in a group, the result of the operation (a * b) is also in the group.
• Associativity: For all a, b, c in the group, (a * b) * c = a * (b * c).
• Identity Element: There is an element (e) in the group such that for all elements g, e * g = g * e = g.
• Inverse Element: For each element g in the group, there is an element g^-1 such that g * g^-1 = g^-1 * g = e.

Group theory lays the foundations for many important areas of mathematics and helps in understanding and constructing more complex structures.

Subgroup Order

The order of a subgroup is an essential concept in understanding the structure of groups. Subgroups themselves are groups formed from a larger group that satisfy all the properties of a group.
The order of a subgroup refers to the number of elements within it and is always a divisor of the order of the whole group (according to Lagrange's Theorem).

• Elements: These are the 'building blocks' of our group. In a subgroup, there will necessarily be fewer elements than in the entire group.
• Lagrange's Theorem: A fundamental theorem that states the order of any subgroup must divide the order of the entire group.

In the context of quotient groups, like \( \mathbb{Z}_{15} / \langle 6 \rangle \), the order of the subgroup \( \langle 6 \rangle \) determines the nature of the quotient group. Here, the number of distinct cosets or the index of the subgroup gives you the quotient group's order, further cementing the ties between subgroup order and larger group structures.