Problem 1 - A Book of Abstract Algebra

Question

In each of the following, \(G\) is a group and \(H\) is a normal subgroup of \(G\). List the elements of \(G / H\) and then write the table of \(G / H\). Example \(G=\mathbb{Z}_{6} \quad\) and \(\quad H=\\{0,3\\}\) The elements of \(G / H\) are the three cosets \(H=H+0=\\{0,3\\}, H+1=\\{1,4\\}\), and \(H+2=\\{2,5\\} .\) (Note that \(H+3\) is the same as \(H+0, H+4\) is the same as \(H+1\), and \(H+5\) is the same as \(H+2\).) The table of \(G / H\) is $$ \begin{array}{c|ccc} \+ & H & H+1 & H+2 \\ \hline H & H & H+1 & H+2 \\ H+1 & H+1 & H+2 & H \\ H+2 & H+2 & H & H+1 \end{array} $$ \(G=\mathbb{Z}_{10}, H=\\{0,5\\} .\left(\right.\) Explain why \(\left.G / H \cong \mathbb{Z}_{5} .\right)\)

Step-by-Step Solution

Answer

The elements of \(G/H\) are \(\{H, H+1, H+2, H+3, H+4\}\). The table of \(G/H\) is like \(\mathbb{Z}_5\), showing \(G/H \cong \mathbb{Z}_5\).

Step 1: Identify the elements of G

The group given is \(G = \mathbb{Z}_{10}\). The elements of \(G\) are \(\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}\).

Step 2: Identify the subgroup H

The normal subgroup given is \(H = \{0, 5\}\).

Step 3: Find the cosets of G/H

The cosets are calculated as follows:\- \(H = H+0 = \{0, 5\}\) \- \(H+1 = \{1, 6\}\) \- \(H+2 = \{2, 7\}\) \- \(H+3 = \{3, 8\}\) \- \(H+4 = \{4, 9\}\).

Step 4: Recognize repeated cosets

Notice that: \- \(H+5 = H\) \- \(H+6 = H+1\) \- \(H+7 = H+2\) \- \(H+8 = H+3\) \- \(H+9 = H+4\). \Thus, all cosets have been identified and no new elements are added.

Step 5: List the elements of G/H

The elements of \(G/H\) are the distinct cosets: \(\{H, H+1, H+2, H+3, H+4\}\).

Step 6: Write the table of G/H

Construct the table of \(G/H\) with the identified cosets: \\[\begin{array}{c|ccccc}\+ & H & H+1 & H+2 & H+3 & H+4 \\hlineH & H & H+1 & H+2 & H+3 & H+4 \H+1 & H+1 & H+2 & H+3 & H+4 & H \H+2 & H+2 & H+3 & H+4 & H & H+1 \H+3 & H+3 & H+4 & H & H+1 & H+2 \H+4 & H+4 & H & H+1 & H+2 & H+3\end{array}\]

Step 7: Explain the isomorphism

\(G/H\) consists of 5 cosets which can be cyclically generated similar to the group \(\mathbb{Z}_5\). Therefore, \(G/H \cong \mathbb{Z}_5\).

Key Concepts

Group Theory

Group theory is a fascinating area of mathematics that deals with groups, which are sets accompanied by an operation satisfying specific axioms like closure, associativity, identity, and invertibility.
Imagine a group as a collection of objects (called elements) where you can combine any two objects and find another object in the same collection. For example, if you add two numbers in the set of integers, the result is also an integer.
To be a group:

• Closure: The group operation on any two elements must yield another element within the group.
• Associativity: The order in which you apply operations doesn't change the outcome. Likewise, for elements \(a, b, \) and \(c\), \((a \, * \, b) \, * \, c = a \, * \, (b \, * \, c)\).
• Identity Element: There must be an element that acts as a neutral element, such that combining it with any element of the group leaves the element unchanged, like 0 for addition.
• Inverse Element: Each element in the group must have an inverse, where combining them together gives the identity element.

Group theory finds applications across the sciences in algebra, number theory, and even crystallography, offering deep insights into the symmetrical properties of objects.

Normal Subgroup

In group theory, a subgroup is a smaller group within a group that itself is a group under the same operation. But what if this subgroup satisfies an even more special property? That's where normal subgroups come in.
A normal subgroup, \(H\), of a group \(G\), is one where any element of the group can be moved across the subgroup without changing the group structure. This means for every element \(g \) in \(G\) and every element \(h \) in \(H\), the equation \(g^{-1} \, h \, g \) also results in an element of \(H\).
Why are they important? Normal subgroups allow us to build quotient groups by forming "cosets," which are critical in understanding how larger groups can be broken down into simpler pieces.

• Denotation: We denote that \(H\) is a normal subgroup of \(G\) as \(H \trianglelefteq G\).
• Usage: They play an essential role in constructing the factor group \(G/H\), helping in studying homomorphisms and isomorphisms.

This makes understanding normal subgroups crucial for grasping the larger framework of group theory.

Cosets

Cosets are a foundational concept used to explore how a group \(G\) can be partitioned using a subgroup \(H\). By considering different ways to "shift" the elements of \(H\) by elements of \(G\), we form subsets of the same size as \(H\).
For any element \(g \) in \(G\), the set \(gH = \{gh \, | \, h \in H\}\) is a coset. Likewise, forms a different partition of the group.
Let's break it down:

• Left Coset: Defined as adding an element \(g\) to each element of \(H\). For example, in the group \(\mathbb{Z}_{6}\), with subgroup \(\{0,3\}\), the left coset for element 1 is \(\{1,4\}\).
• Importance: Cosets help in forming the quotient group \(G/H\). Where distinct cosets become single elements.

By examining cosets, we gain insights into the structure and simplicity of larger groups, breaking them down into more manageable parts.

Group Isomorphism

In group theory, saying two groups are isomorphic is like saying two things are identical in structure but not necessarily in appearance. A group isomorphism states there exists a one-to-one correspondence between the elements of two groups such that the group operation is preserved.
Imagine isomorphisms as a perfect translation between two different languages where meaning is preserved.

• Definition: A function \(f: G \to K\) is a group isomorphism if it is a bijection (one-to-one and onto) and respects the group operation, so \(f(x \, * \, y) = f(x) \, *' \, f(y)\) for all elements \(x, y\) in \(G\).
• Notation: We often write \(G \cong K\) to denote that groups \(G\) and \(K\) are isomorphic.
• Example: In our exercise, \(G/H\) is isomorphic to \(\mathbb{Z}_5\) because their structures, like being cyclic, align perfectly.

Group isomorphism enables us to simplify complex group behaviors by relating them to groups we already understand well, hence they play a vital role in classification and simplifying group studies.