Question

Let \(\phi: G \rightarrow G^{\prime}\) be a homomorphism, \(K=\) Kern \(\phi\), and \(a \in G\). Show that \(\\{x \in G\) \(\phi(x)=\phi(a)\\}=a K,\) the left coset of \(K\) to which the element \(a\) belongs.

Short answer

The set \( \{x \in G \mid \phi(x) = \phi(a)\} \) is equal to \( aK \).

Step-by-step solution

Define the left coset

The left coset of a subgroup \( K \) in \( G \) with respect to an element \( a \) of \( G \) is defined as the set of all elements of the form \( ak \) where \( k \in K \). In our problem, this is denoted as \( aK \).

Consider any element in the set \( \{x \in G \mid \phi(x) = \phi(a)\} \)

Take an arbitrary element \( x \in G \) such that \( \phi(x) = \phi(a) \). We want to show that this implies \( x \in aK \).

Use properties of homomorphism to equate images

Since \( \phi \) is a homomorphism and \( \phi(x) = \phi(a) \), we have: \( \phi(x) = \phi(a) = \phi(a)x = \phi(x)\).

Rearrange to show element belongs to coset

The equation \( \phi(xa^{-1}) = e' \) follows because \( \phi \) is a homomorphism, and the identity element in \( G' \) is \( e' \). This implies that \( xa^{-1} \in K \) because \( K = \text{Kern }\phi = \{ g \in G \mid \phi(g) = e' \} \). Therefore, \( x = ak \) for some \( k \in K \) which shows \( x \in aK \).

Consider elements of the coset \(aK\)

Now consider any element \( y = ak \) where \( k \in K \). Then \( \phi(y) = \phi(ak) = \phi(a)\phi(k) = \phi(a)e' = \phi(a) \). Thus, any element in \( aK \) satisfies \( \phi(y) = \phi(a)\).

Conclude equality of sets

Since we have shown that: 1) any element \( x \) with \( \phi(x) = \phi(a) \) is in \( aK \), and 2) any element in \( aK \) has the property that \( \phi(y) = \phi(a) \), it follows that the set \( \{x \in G \mid \phi(x) = \phi(a)\} \) is equal to the left coset \( aK \).

Key concepts

Cosets

In group theory, a **coset** is a form of subgroup that is defined in relation to an element from the original group. More specifically, a left coset of a subgroup \( K \) in group \( G \) related to an element \( a \) in \( G \) is the set of all products of \( a \) with each element of \( K \).
For instance, if \( K \) is a subgroup of \( G \), then the left coset of \( K \) by \( a \) is \( aK = \{ ak \mid k \in K \} \). The idea of a coset is useful for understanding how groups can be partitioned and how subgroups relate to their parent groups.
Understanding cosets can help simplify complex group structures and analyze homomorphisms. Cosets play a significant role when demonstrating that particular properties hold over the whole group, especially when examining normal subgroups and factor groups.

Kernels

The concept of a **kernel** is critical in understanding group homomorphisms. The kernel of a homomorphism \( \phi: G \rightarrow G^{\prime} \) is the set of elements in \( G \) that map to the identity element in \( G^{\prime} \).
The kernel, often denoted as \( \text{Kern } \phi \), forms a normal subgroup of \( G \). It contains all the elements that do not "change" when the homomorphism is applied, mathematically described as \( \text{Kern } \phi = \{ g \in G \mid \phi(g) = e' \} \), where \( e' \) is the identity in \( G^{\prime} \).
This property is instrumental since it tells us about the structure-preserving nature of homomorphisms and can be used to test if a homomorphism is injective (one-to-one): a homomorphism is injective if and only if its kernel is trivial, meaning it only contains the identity element from \( G \).

Group Theory

**Group theory** is a branch of mathematics that studies algebraic structures known as groups, which are sets equipped with an operation that combines two elements to form a third element. The operation must satisfy four key properties: closure, associativity, the presence of an identity element, and the existence of inverses for each element.
Group theory explores these initial properties to classify groups, understand their substructures, and determine how they function under various conditions. Groups serve as the foundation for many areas of mathematics and have applications in fields as diverse as physics, cryptography, and chemistry.

• Closure: If \( a \) and \( b \) are in group \( G \), then the result of the operation, \( ab \), is also in \( G \).
• Associativity: For any \( a, b, c \in G \), the equation \( (ab)c = a(bc) \) holds.
• Identity: There exists an element \( e \in G \) such that for every element \( a \in G \), \( ae = ea = a \).
• Inverse: For each \( a \in G \), there exists an element \( b \in G \) such that \( ab = ba = e \).

Grasping group theory basics paves the way for deeper mathematical exploration and problem-solving in homogeneous environments.

Subgroups

A **subgroup** is a group contained within a larger group that maintains the structure and properties of a group itself. For a subset \( H \) of a group \( G \) to be considered a subgroup, it must satisfy the four group properties: closure, associativity, identity, and inverses.
When analyzing subgroups, there are significant types to consider:

• Normal Subgroups: These are subgroups \( N \subset G \) where every left coset aN is also a right coset Na. Normal subgroups are crucial for creating quotient groups.
• Proper Subgroups: These are subgroups \( H \) that are strictly smaller than \( G \) (i.e., \( H eq G \)).
• Trivial Subgroup: This subgroup contains only the identity element.

Identifying subgroups helps understand the symmetry and structure within a group. They are fundamental in decomposing complicated groups into simpler components, aiding in analysis and solution strategies.