Question

Let \(H=5 \mathbb{Z}\) in \(\mathbb{Z}\). Determine whether the following cosets of \(H\) are the same: (a) \(12+H\) and \(27+H\) (b) \(13+H\) and \(-2+H\) (c) \(126+H\) and \(-1+H\)

Short answer

(a) Same coset, (b) Same coset, (c) Different coset.

Step-by-step solution

Define Cosets in the Group

A coset \(a + H\) in the group \(\mathbb{Z}\) is defined as the set \( \{ a + h \mid h \in H \}\). In our case, since \(H = 5\mathbb{Z}\), the elements of \(H\) are all multiples of 5. Therefore, the coset \(a + H\) is \( \{ a + 5k \mid k \in \mathbb{Z} \} \). Cosets \(a+H\) and \(b+H\) are the same if \(a \equiv b \pmod{5}\).

Check Equivalence for (a)

To determine if \(12+H\) and \(27+H\) are the same, check if \(12 \equiv 27 \pmod{5}\). Calculate: \(27 - 12 = 15\). Since \(15\) is divisible by 5, \(12 \equiv 27 \pmod{5}\). Therefore, \(12+H\) and \(27+H\) are the same coset.

Check Equivalence for (b)

To determine if \(13+H\) and \(-2+H\) are the same, check if \(13 \equiv -2 \pmod{5}\). Calculate: \(13 - (-2) = 15\). Since \(15\) is divisible by 5, \(13 \equiv -2 \pmod{5}\). Therefore, \(13+H\) and \(-2+H\) are the same coset.

Check Equivalence for (c)

To determine if \(126+H\) and \(-1+H\) are the same, check if \(126 \equiv -1 \pmod{5}\). Calculate: \(126 - (-1) = 127\). Calculate the remainder: \(127 \div 5 = 25\) remainder \(2\), so \(126 \equiv 2 \pmod{5}\) and \(-1 \equiv 4 \pmod{5}\). Since \(2 ot\equiv 4 \pmod{5}\), \(126+H\) and \(-1+H\) are not the same coset.

Key concepts

Understanding Cosets

Cosets are an essential concept in group theory, especially when dealing with the structure of groups like the integers \(\mathbb{Z}\). A coset \(a + H\) of a subgroup \(H\) in a group \(G\) is defined as the set of elements obtained by adding each element of \(H\) to a fixed element \(a\) from the group. In simple terms, it's like moving the entire subgroup by a fixed amount. This concept helps in partitioning a group into disjoint subsets, each of which is a coset.One of the pivotal properties of cosets is that two cosets \(a+H\) and \(b+H\) are identical if the difference between \(a\) and \(b\) is an element of \(H\). More mathematically, \(a\) and \(b\) lie in the same coset if \(a - b\) belongs to \(H\).

• Cosets demonstrate how a whole group can be divided into smaller, equally sized entities.
• Investigating cosets helps understand quotient groups, which describe the group structure more comprehensively.

Cosets thus serve as a fundamental tool in understanding the internal architecture of a group and are key in exploring more complex concepts like normal subgroups and Lagrange's theorem.

Exploring Modulo Arithmetic

Modulo arithmetic is like cutting numbers down to size, only caring about their remainders when divided by a certain number. Imagine you have a clock where the hours wrap around after hitting 12. Modulo arithmetic works similarly. You wrap around once you reach a certain number, called the modulus.For example, in the integer group \(\mathbb{Z}\), if we are using modulus 5, we only worry about the remainder when numbers divide by 5. So, the statement \(12 \equiv 2 \pmod{5}\) tells us simply that when 12 is divided by 5, the remainder is 2.In the context of cosets, modulo arithmetic helps determine if two cosets are equal. If the difference between the representational numbers of the two cosets is divisible by the given modulus, they are the same. In other words, \(a + H\) equals \(b + H\) if \(a \equiv b \pmod{n}\) where \(H = n\mathbb{Z}\).

• Modulo arithmetic is powerful in exploring periodic or cyclic structures.
• It's used in a multitude of areas like cryptography, computer science, and more.
• Understanding modulo arithmetic offers insight into how numbers relate within different modular systems.

With its simplicity and utility, modulo arithmetic is invaluable in both theoretical and applied mathematics.

Equivalence Relations in Group Theory

An equivalence relation is a way of grouping objects within a set by a relation that satisfies three properties: reflexivity, symmetry, and transitivity. In simpler terms:

• Reflexivity means every element is related to itself.
• Symmetry means that if one element is related to a second element, then the second is related to the first.
• Transitivity means that if one element is related to a second, and the second to a third, then the first is related to the third.

In group theory, equivalence relations help define how elements in a set can be grouped into what we call equivalence classes. For instance, in the integer group \(\mathbb{Z}\) with a module like 5, numbers that share the same remainder when divided by 5 belong to the same equivalence class or coset.Every equivalence class forms a partition of the set, ensuring that each element belongs to one and only one class, which is crucial in understanding symmetrical structures within groups. Understanding equivalence relations helps grasp why certain elements "look the same" when analyzed under specific rules or viewpoints, like in modular systems and coset constructions.