Problem 2 - A Book of Abstract Algebra

Question

In each of the following, \(H\) is a subset of \(\mathbb{R} \times \mathbb{R}\) (a) Prove that \(H\) is a normal subgroup of \(\mathbb{R} \times \mathbb{R}\). (Remember that every subgroup of an abelian group is normal.) (b) In geometrical terms, describe the elements of the quotient group \(G / H\). (c) In geometrical terms or otherwise, describe the operation of \(G / H\). $$ H=\\{(x, y): y=-x\\} $$

Step-by-Step Solution

Answer

(a) H is a normal subgroup of \(G\). (b) Elements of \(G/H\) are lines parallel to \(y = x\). (c) The operation in \(G/H\) is addition of these lines.

Step 1: Understand the Setup

We have a group defined as \(G = \mathbb{R} \times \mathbb{R}\), where each element is a pair \((x, y)\). The subset \(H\) is defined by the set of pairs \((x, y)\) such that \(y = -x\). We need to examine \(H\) within the context of the group \(G\).

Step 2: Prove H is a Subgroup (and Hence Normal)

To prove \(H\) is a subgroup, check closure under group operation (addition) and inverses. If \((x_1, y_1)\) and \((x_2, y_2)\) are in \(H\), then \(y_1 = -x_1\) and \(y_2 = -x_2\). Thus, \[(x_1, y_1) + (x_2, y_2) = (x_1 + x_2, y_1 + y_2) = (x_1 + x_2, -x_1 - x_2),\] which is in \(H\). Inverses: \[-(x, y) = (-x, -y) = (-x, x),\] which is in \(H\). As \(G\) is abelian, all subgroups are normal.

Step 3: Describe Elements of Quotient Group G/H

Each element in the quotient group \(G/H\) is a coset of \(H\) in \(G\). For an element \((a, b) \in G\), the coset is \((a, b) + H = \{(a+x, b-x) | x \in \mathbb{R}\}\). Geometrically, these cosets are lines parallel to \(y = x + b - a\).

Step 4: Describe the Operation in G/H

In \(G/H\), the operation is addition of cosets. For cosets represented by \((a, b) + H\) and \((c, d) + H\), their sum is the coset \((a + c, b + d) + H\). This results in another line parallel to \(y = x\), shifted by the vector sum.

Key Concepts

Abelian Group

An Abelian group refers to a group where the group operation is commutative, which means that the order of elements does not affect the result of an operation. In mathematical terms, a group \(G\) is Abelian if for all elements \(a, b \in G,\) it holds that \(a + b = b + a\). This clear and straightforward property significantly simplifies many aspects of group theory because it guarantees that the structure remains consistent regardless of the sequence of operations applied.

• Example: A common example of an Abelian group is the integers under addition, \((\mathbb{Z}, +)\), because for any two integers \(a\) and \(b\), it is always true that \(a + b = b + a\).
• Application: In an Abelian group, every subgroup is normal. This means that cosets formed via any subgroup are well-behaved, which is a great advantage when analyzing the group's structure.
• Relevance: The group \( \mathbb{R} \times \mathbb{R} \) is Abelian since both components, \(\mathbb{R},\) are individually Abelian. This ensures that any subgroup like \(H\) that is defined over this group will also be normal by default.

Quotient Group

A quotient group arises when we partition a group into subsets known as cosets of a subgroup. Given a group \(G\) and a normal subgroup \(H\), the set of cosets forms a new group called the quotient group, denoted as \(G/H\). The elements of this quotient group are these cosets, and the group operation on them is defined in terms of the original group operation on \(G\).

• Structure: The cosets are essentially the distinct orbits of a normal subgroup \(H\) in \(G\), reflecting equivalence classes created by \(H\).
• Example: For \(G = \mathbb{R} \times \mathbb{R}\) and \(H = \{(x, y): y = -x\}\), each element of \(G/H\) can be identified with lines parallel to certain sets in the ambient space.
• Insight: By using quotient groups, complex group operations can be reduced to simpler analyses of these smaller, more manageable pieces.

Cosets

Cosets are fundamental to understanding the structure of groups, especially when dealing with subgroups. If \(H\) is a subgroup of a group \(G\), a left coset is formed by taking an element \(g \in G\) and applying the operation with every element in \(H\). This results in a set \(gH = \{gh : h \in H\}\). A right coset is similarly defined as \(Hg = \{hg : h \in H\}\). In an Abelian group, these are always equal.
Cosets play a critical role in forming quotient groups by providing the building blocks for these more abstract constructions.

• Properties: In our example, with \(H = \{(x, y) : y = -x\}\), any pair \((a, b)\) in \(G\) leads to a left coset \((a, b) + H = \{(a + x, b - x) : x \in \mathbb{R}\}\).
• Geometry: Geometrically, each coset represents a line in the plane with a specific orientation, making it easier to visualize their arrangement and operation.
• Usage: Cosets help establish equivalence relations that simplify group structures and operations over them, laying the groundwork for more advanced analysis.

Group Operation

The group operation is the rule according to which elements from a group combine. In the context of the Abelian group \(\mathbb{R} \times \mathbb{R}\), this operation is vector addition. For any two elements \((x_1, y_1)\) and \((x_2, y_2)\) in the group, their sum would be \((x_1 + x_2, y_1 + y_2)\).
This operation extends naturally to the quotient group \(G/H\) by operating on entire cosets. When you "add" two cosets \((a, b) + H\) and \((c, d) + H\), their sum in the quotient group is a new coset \((a + c, b + d) + H\).

• Consistency: The operation maintains the structure and orientation of lines represented by cosets, ensuring a coherent group structure.
• Visual: This is akin to sliding lines in parallel until they align under a specific rule dictated by element addition, creating a new configuration within \(G/H\).
• Simplification: Such operations allow us to derive meaningful insights into group behavior and simplify the understanding of complex algebraic structures.