Question

Determine whether the indicated subgroup is normal in the indicated group. $$ 3 \mathbb{Z} \text { in Z } $$

Short answer

Yes, the subgroup \\(3Z\\) is normal in \\(Z\\).

Step-by-step solution

Identify the Groups

Here, we need to identify the group and the subgroup. The group is the set of all integers, \( Z \), and the subgroup is \( 3Z \), which consists of all multiples of 3.

Define Normal Subgroup

A subgroup \( H \) of a group \( G \) is normal if for every element \( g \) in \( G \), the left coset \( gH \) is equal to the right coset \( Hg \).

Check Cosets Equality

In a commutative (abelian) group such as \( Z \), left and right cosets are always equal. Thus, \( g + 3Z = 3Z + g \) holds for every integer \( g \).

Conclusion

Since \( Z \) is an abelian group, \( 3Z \) is normal in \( Z \) because all subgroups of an abelian group are normal.

Key concepts

Abelian Group

An abelian group, named after the mathematician Niels Henrik Abel, is a fundamental concept in group theory. In an abelian group, the group operation is commutative. This means that the order in which you combine elements does not affect the outcome. If you have any two elements \( a \) and \( b \) in an abelian group \( G \), the equation \( a * b = b * a \) always holds true.

Abelian groups are important because they provide a simplified structure to study, as commutativity allows for easier manipulation and understanding of group properties. Examples of abelian groups include:

• The group of integers under addition \( \mathbb{Z} \).
• Any vector space under vector addition.
• The group of real numbers excluding zero, under multiplication.

Understanding that \( \mathbb{Z} \), the group of integers, is abelian helps to easily ascertain properties like the normality of subgroups.

Integers Group

The integers group, often denoted as \( \mathbb{Z} \), is a classic example used in group theory to illustrate key concepts. This group consists of all positive and negative whole numbers, including zero, and the operation used is addition.

The properties of this group include:

• Closure: The sum of any two integers is an integer.
• Associativity: Addition is associative, meaning \( (a + b) + c = a + (b + c) \).
• Identity Element: The identity element is 0, as adding 0 to any integer returns the same integer.
• Inverse Element: Each integer \( a \) has an inverse \( -a \), which when added results in the identity, \( a + (-a) = 0 \).

These properties confirm that \( \mathbb{Z} \) forms a group under addition, specifically an abelian group, given its commutative property.

Cosets

Cosets are a way of exploring the structure of a group using its subgroup. Given a group \( G \) and a subgroup \( H \), a left coset of \( H \) in \( G \) is formed by taking an element \( g \) from \( G \) and combining it with every element of \( H \) through the group operation.

Mathematically, the left coset of \( H \) in \( G \) with respect to \( g \) is written as \( gH = \{ g + h \mid h \in H \} \). Similarly, right cosets are defined as \( Hg = \{ h + g \mid h \in H \} \).

In abelian groups such as \( \mathbb{Z} \), left and right cosets are always the same, hence \( g + 3\mathbb{Z} = 3\mathbb{Z} + g \), confirming that the subgroup \( 3\mathbb{Z} \) is normal in \( \mathbb{Z} \).

Cosets provide insight into partitioning groups into equivalent classes and understanding the quotient group structure.

Group Theory

Group theory forms the backbone of modern algebra and is used to study algebraic structures known as groups. It is a mathematical field that investigates the symmetry and structure found within groups.

In group theory, we define a group as a set, \( G \), combined with an operation satisfying four fundamental properties:

• Closure
• Associativity
• Identity
• Invertibility

These criteria ensure that groups can be rigorously studied and their structures understood.

One critical aspect of group theory is determining if a subgroup is a normal subgroup. A subgroup \( H \) of \( G \) is normal if for every element \( g \) in \( G \), the left coset \( gH \) equals the right coset \( Hg \). Normal subgroups lead to the construction of quotient groups, essential tools in decomposing groups into simpler parts.

Group theory's applications span across mathematics and other sciences, including physics, chemistry, and computer science, showcasing its foundational importance.