Question

Let \(H\) and \(K\) be subgroups of a group \(G\) and for all \(a, b \in G\) let \(a \sim b\) if and only if \(a=h b k\) for some \(h \in H\) and \(k \in K\). Show that the relation \(\sim\) so defined is an equivalence relation. Describe the equivalence classes (which are called double cosets).

Short answer

The relation \(\sim\) is an equivalence relation on \(G\). Double cosets are of the form \(HbK\).

Step-by-step solution

Show Reflexivity

To prove reflexivity, we need to show that for any element \(a \in G\), \(a \sim a\) holds. Since \(H\) and \(K\) are subgroups of \(G\), they both contain the identity element, denoted by \(e\). Therefore, for any \(a \in G\), we have \(a = eae\), where \(e \in H\) and \(e \in K\). Hence, \(a \sim a\), establishing reflexivity.

Show Symmetry

For symmetry, we need to show that if \(a \sim b\), then \(b \sim a\) for any elements \(a, b \in G\). By definition, if \(a \sim b\), there exist \(h \in H\) and \(k \in K\) such that \(a = hbk\). Taking inverses, we get \(b = h^{-1}a k^{-1}\), since inverses of \(h\) and \(k\) are also in \(H\) and \(K\) respectively. Thus, \(b \sim a\), establishing symmetry.

Show Transitivity

To establish transitivity, assume \(a \sim b\) and \(b \sim c\). Then, there exist elements \(h_1, h_2 \in H\) and \(k_1, k_2 \in K\) such that \(a = h_1bk_1\) and \(b = h_2ck_2\). Substituting the second expression into the first gives us \(a = h_1(h_2ck_2)k_1 = (h_1h_2)c(k_2k_1)\). Since \(h_1h_2 \in H\) and \(k_2k_1 \in K\), we have \(a \sim c\), proving transitivity.

Describe the Equivalence Classes (Double Cosets)

Each equivalence class, or double coset, is described as \(Hbk\) for some element \(b \in G\). Explicitly, a double coset is the set of all elements of the form \(hbk\) where \(h \in H\) and \(k \in K\). Essentially, the double coset \(HbK\) collects all elements from \(G\) that are \(H\)-related to any element in \(bK\).

Key concepts

Subgroups

In group theory, a subgroup is essentially a smaller group contained within a larger group. To qualify as a subgroup, a subset must satisfy two main conditions: closure and the presence of inverses. Here's a breakdown:

• Closure: If you take any two elements from the subset and combine them using the group operation, the result should still be in the subset.
• Inverses and Identity: Every element in the subset must have an inverse, and importantly, the subset must contain the identity element of the larger group.

Subgroups are significant because they help us understand the structure of the larger group. For instance, if we consider a group of symmetries of a square, subgroups could represent symmetries like rotations or reflections.
Exploring the concept of subgroups allows mathematicians to study properties like symmetry and invariant factors in more complex groups. They can construct smaller, more manageable systems while staying within the scope of the original group. This simplification is crucial in many theoretical and practical applications in mathematics.

Group Theory

Group theory is a branch of mathematics that studies groups, which are sets equipped with an operation that combines any two of its elements to form a third element. The cool part here is this operation should satisfy four basic properties:

• Closure: The result of combining two elements is also an element of the same set.
• Associativity: The way you group the elements when combining them doesn't change the result. For example, \( (a imes b) imes c = a imes (b imes c) \).
• Identity Element: There is an element which, when used in the operation with any other element in the group, results in the other element. This element is called the identity.
• Inverse Element: For every element in the group, there should be another element which, when combined with the first, produces the identity element.

Group theory has widespread applications in various fields, including physics, chemistry, and computer science. For example, it's used to study symmetry in molecules, solve Rubik's cubes, and even understand fundamental particles in physics. It provides a powerful language for discussing and solving problems involving symmetry and transformations.

Double Cosets

In the world of groups, double cosets play a fascinating role in understanding symmetrical and structural properties of the group. Imagine we have two subgroups, say \( H \) and \( K \) of a group \( G \). A double coset is a "slice" of the group \( G \) formed by using these two subgroups.
Double cosets are formed as follows: for a given element \( b \) in \( G \), the double coset is the set of elements of the form \( hbk \), where \( h \) is from \( H \) and \( k \) is from \( K \). This gives us the double coset \( HbK \).
What's the importance of double cosets?

• Partition of the Group: Double cosets partition the group into distinct, non-overlapping sections based on the subgroups used to create them.
• Symmetric Analysis: They help in analyzing the symmetry and structure of the group by understanding how it can be divided up using its subgroups.
• Applications: Double cosets are used in mathematical fields like geometry and physics to understand symmetrical operations in space and physical transformations.

By using double cosets, mathematicians can greatly simplify complex problems about the structure and types of transformations a set can undergo, making it a fundamental tool in the toolkit of anyone studying group theory.