Problem 3

Question

Let $H$ and $K$ be normal subgroups of a group $G$, with $H \subseteq K$. Define $\phi: G / H \rightarrow G / K$ by: $\phi(H a)=K a$. Prove the following : $$ \phi \text { is suriective. } $$

Step-by-Step Solution

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Answer

The map $\phi: G/H \rightarrow G/K$ is surjective because for every $Ka \in G/K$, $\phi(Ha) = Ka$.

1Step 1: Understanding Surjectivity

To prove that a function is surjective, we need to show that for every element in the codomain, there is a preimage in the domain. In our case, for each element $Kb$ in $G/K$, there must exist an element $Ha$ in $G/H$ such that $\phi(Ha) = Kb$.

2Step 2: Define the Function

Recall the function $\phi(G/H \rightarrow G/K)$ defined by $\phi(Ha) = Ka$. $Ha$ is a coset in $G/H$ and $Ka$ is a coset in $G/K$.

3Step 3: Surjectivity Condition

To show $\phi$ is surjective, consider any element $Ka$ in $G/K$. The goal is to find some element $a'$ in $G$ such that $\phi(Ha') = Ka$.

4Step 4: Use the Definition of $\phi$

By definition, for any element $a$ in $G$, $\phi(Ha) = Ka$. This means $\phi$ sends the coset $Ha$ in $G/H$ directly to $Ka$ in $G/K$. Therefore, for any $Ka$ in $G/K$, $a$ itself already serves as the preimage under $\phi$, namely $a' = a$.

5Step 5: Concluding Surjectivity

Since for every $Ka$ there is a specific $Ha$ such that $\phi(Ha) = Ka$, we conclude that the function $\phi: G/H \rightarrow G/K$ is indeed surjective.

Key Concepts

Normal SubgroupsSurjective FunctionCosetsHomomorphism

Normal Subgroups

In group theory, normal subgroups play a crucial role in understanding the structure of groups. A subgroup, denoted as $H$, of a group $G$, is normal if it is invariant under conjugation by any element of $G$. This means for every element $g \in G$ and every element $h \in H$, the element $g h g^{-1} $ is still in $H$.

Normal subgroups are significant because they allow us to form quotient groups. A subgroup $H$ is said to be a normal subgroup of $G$, written as $H \trianglelefteq G$, if the left and right cosets of $H$ in $G$ are the same.

Normal subgroups enable the construction of simpler groups, known as quotient groups, from more complex groups.
They help in determining homomorphisms between groups.

Understanding normal subgroups is critical as they facilitate various concepts in group theory, like the definition of homomorphisms and cosets.

Surjective Function

A function is surjective, or onto, if every element in the function’s codomain is an image of at least one element from the domain. This means for a function $f: A \rightarrow B$ to be surjective, for every $b \in B$, there exists some $a \in A$ such that $f(a) = b$.

In the context of group theory, surjectivity ensures that the function relates every element of the codomain back to an element of the domain. This property was crucial in the exercise involving the function $\phi: G/H \rightarrow G/K$, where surjectivity had to be proved.

Surjective functions cover the entire codomain, leaving no element unmapped.
Proving surjectivity often involves showing that arbitrary elements of the codomain can be reached from the domain.

Surjective functions are vital in group theory when demonstrating that mappings cover the entire range set.

Cosets

Cosets are a fundamental concept in group theory used to build quotient structures. When considering a subgroup $H$ of a group $G$, the left coset of $H$ containing an element $g \in G$ is denoted by $gH$ and is the set $ \{gh \mid h \in H\} $. Similarly, the right coset is $Hg = \{hg \mid h \in H\}$.

Cosets partition the group into equal-sized subsets, and every element of the group $G$ is contained in exactly one coset of $H$. Cosets are used to form quotient groups like $G/H$, which help in analyzing the structure of $G$.

Cosets are useful in understanding normal subgroups and homomorphisms.
They help in visualizing the division of groups into smaller, more manageable parts.

In our exercise, the function $\phi(Ha) = Ka$ was established by using cosets from $G/H$ and $G/K$. This mapping was important to show how elements from one quotient group could map onto another, particularly focusing on surjectivity.

Homomorphism

A homomorphism is a structure-preserving map between two algebraic structures, such as groups. In group theory, a homomorphism $\phi: G \rightarrow H$ satisfies the property that for any elements $a, b \in G$, $\phi(ab) = \phi(a)\phi(b)$. This property ensures that the operation within the domain is preserved in the codomain.

Homomorphisms are essential for studying the relationships between different groups. The kernel of a homomorphism, which is the preimage of the group identity in the codomain, often reveals a great deal of information about the structure of the group.

Homomorphisms help identify group properties like isomorphisms.
They enable the analysis of group structures by preserving operations.

In the given exercise, although the direct notion of homomorphism wasn't explicitly used, the concept of mapping between cosets aligns closely with how homomorphisms function to link structures in group theory.

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