Question

Let \(H\) be a subgroup of a group \(G\). For any \(a, b \in G,\) let \(a \sim b\) if and only if \(a b^{-1} \in H\). Show that the relation \(\sim\) so defined is an equivalence relation on \(G,\) with equivalence classes the right cosets \(\mathrm{Ha}\) of \(\mathrm{H}\).

Short answer

The relation \(\sim\) is an equivalence relation with equivalence classes as the right cosets of \(H\) in \(G\).

Step-by-step solution

Understanding Equivalence Relations

To show that \(\sim\) is an equivalence relation, we need to demonstrate that it satisfies three properties: reflexivity, symmetry, and transitivity.

Proving Reflexivity

For reflexivity, for any given element \(a \in G\), we need to check if \(a \sim a\). We compute:\[ a a^{-1} = e \in H, \]where \(e\) is the identity element of \(G\). Since \(e\) is in \(H\) (identity must be in every subgroup), \(a \sim a\) is true, satisfying reflexivity.

Proving Symmetry

For symmetry, assume \(a \sim b\) for some \(a, b \in G\). This implies:\[ a b^{-1} \in H. \]We need to show \(b \sim a\), meaning \(b a^{-1} \in H\). Since \(H\) is a subgroup, it is closed under taking inverses:\[ (a b^{-1})^{-1} = b a^{-1} \in H. \]Thus, \(b \sim a\), satisfying symmetry.

Proving Transitivity

For transitivity, assume \(a \sim b\) and \(b \sim c\). This means:\[ a b^{-1} \in H \quad \text{and} \quad b c^{-1} \in H. \]We need to show \(a \sim c\), meaning \(a c^{-1} \in H\). Notice:\[ a c^{-1} = (a b^{-1})(b c^{-1}) \in H \text{ because } a b^{-1}, b c^{-1} \in H \text{ and } H \text{ is closed under multiplication.} \]Thus, \(a \sim c\), satisfying transitivity.

Identifying Equivalence Classes

The equivalence class of an element \(a \in G\) is the set of all elements \(b \in G\) such that \(a \sim b\), meaning \(a b^{-1} \in H\). This can be rewritten as: \[ b = ha \text{ for some } h \in H, \]which is precisely the right coset \(Ha\). Thus, the equivalence classes are exactly the right cosets of \(H\) in \(G\).

Key concepts

Group Theory

In mathematics, group theory is a fascinating field that studies algebraic structures known as groups. A group is a set equipped with a single binary operation that combines any two elements to form a third element in such a way that four primary conditions are met: closure, associativity, identity, and the existence of inverse elements. One of the simplest examples of a group is the set of integers with the operation of addition.

• Closure: For all elements \(a, b\) in the group, the result of the operation \(a \cdot b\) is also in the group.
• Associativity: For all \(a, b, c\) in the group, \((a \cdot b) \cdot c = a \cdot (b \cdot c)\).
• Identity: There exists an element \(e\) in the group such that for every element \(a\), \(e \cdot a = a \cdot e = a\).
• Inverse: For each element \(a\) in the group, there exists an element \(b\) such that \(a \cdot b = b \cdot a = e\).

Group theory is essential in understanding various algebraic structures and is widely used in fields like physics, chemistry, and cryptography. Groups can be finite or infinite, and some groups have additional properties like being abelian, where the order of operation doesn't matter.

Cosets

Cosets are particularly important in the study of groups and subgroups. Given a group \(G\) and a subgroup \(H\), we form cosets by taking an element from \(G\) and combining it with all elements of \(H\). If \(a\) is an element in \(G\), then:

• Left coset: The set \(aH = \{ah : h \in H\}\), where each element of \(H\) is multiplied on the left by \(a\).
• Right coset: The set \(Ha = \{ha : h \in H\}\), where each element of \(H\) is multiplied on the right by \(a\).

These cosets partition the group \(G\) into disjoint, equal-sized subsets, meaning that any two cosets either do not overlap or are identical. The concept of cosets is crucial for understanding the structure of groups and plays a vital role in the Lagrange's theorem, which connects the number of cosets with the orders of the group and its subgroups.

Subgroups

A subgroup is essentially a smaller group within a larger group \(G\) that itself satisfies the group properties. If \(H\) is a subset of \(G\), we call \(H\) a subgroup if it fulfills the following conditions:

• Closure: For every \(a, b\) in \(H\), the product \(a \cdot b\) is also in \(H\).
• Identity: The identity element of \(G\) is also in \(H\).
• Inverses: For every \(a\) in \(H\), its inverse \(a^{-1}\) is also in \(H\).

Subgroups provide insight into the internal symmetry and structure of groups. The concept of subgroups aids in exploring equivalence relations where elements are related if they are in the same coset of the subgroup. This understanding is fundamental in various branches of mathematics and helps in the categorization and solution of more complex algebraic problems.