Problem 39 - A First Course in Abstract Algebra

Question

Let \(G\) and \(G^{\prime}\) be groups, and let \(H\) and \(H^{\prime}\) be normal subgroups of \(G\) and \(G^{\prime}\), respectively. Let \(\phi\) be a homomorphism of \(G\) into \(G^{\prime}\). Show that \(\phi\) induces a natural homomorphism \(\phi_{n}:(G / H) \rightarrow\left(G^{\prime} / H^{\prime}\right)\) if \(\phi[H] \subseteq\) \(H^{\prime}\), (This fact is used constantly in algebraic topology.)

Step-by-Step Solution

Answer

\(\phi\) induces \(\phi_n: (G/H) \to (G'/H')\) through cosets as long as \(\phi[H] \subseteq H'\).

Step 1: Define the Quotient Groups

Define the quotient groups \(G / H\) and \(G' / H'\). These groups consist of the cosets of \(H\) in \(G\) and \(H'\) in \(G'\), respectively. For instance, an element of \(G / H\) is of the form \(gH = \{gh \mid h \in H\}\).

Step 2: Define Natural Homomorphism \(\phi_n\)

We construct a homomorphism \(\phi_n: G/H \to G'/H'\) such that for any coset \(gH\) in \(G/H\), \(\phi_n(gH) = \phi(g)H'\). This is well-defined if \(\phi[H] \subseteq H'\), ensuring that different representatives of cosets map to the same coset.

Step 3: Verify Well-Definedness of \(\phi_n\)

To check that \(\phi_n\) is well-defined, take any two elements \(g_1, g_2 \in G\) such that \(g_1H = g_2H\). This implies \(g_2 = g_1h\) for some \(h \in H\). The image under \(\phi\) is \(\phi(g_2) = \phi(g_1)\phi(h)\), and since \(\phi(h) \in H'\), the coset \(\phi(g_2)H' = \phi(g_1)\phi(h)H' = \phi(g_1)H'\), confirming well-definedness.

Step 4: Show \(\phi_n\) is a Homomorphism

Consider any two cosets \(g_1H\) and \(g_2H\) in \(G/H\). We need to show \(\phi_n((g_1g_2)H) = \phi_n(g_1H)\phi_n(g_2H)\). This follows since \(\phi_n((g_1g_2)H) = \phi(g_1g_2)H' = \phi(g_1)\phi(g_2)H'\), and \(\phi_n(g_1H)\phi_n(g_2H) = \phi(g_1)H' \cdot \phi(g_2)H' = \phi(g_1)\phi(g_2)H'\), proving that \(\phi_n\) is a homomorphism.

Step 5: Show \(\phi_n\) is Induced by \(\phi\)

Since \(\phi_n(gH) = \phi(g)H'\), it is clear that \(\phi_n\) is constructed directly from \(\phi\). The role of normality is important because \(H\) being normal in \(G\) and \(H'\) in \(G'\) ensures that the cosets are closed under group operations. Thus, \(\phi_n\) is a natural homomorphism induced by \(\phi\).

Key Concepts

Normal Subgroups

A subgroup \( H \) of a group \( G \) is termed "normal" if it satisfies certain symmetrical conditions in relation to the group operation. This means for any element \( g \) in \( G \), the left and right cosets \( gH \) and \( Hg \) must be exactly the same, i.e., \( gH = Hg \). When a subgroup has this property, it allows for certain algebraic constructions, like quotient groups, to be made possible.

• Normal subgroups are denoted as \( H \trianglelefteq G \).
• They enable us to partition a group into non-overlapping "cosets."
• Crucially, they allow operations on these cosets to be well-defined, since no matter which representative you start with, operations within the group give consistent results.

Understanding normal subgroups is essential because they lay the groundwork for defining quotient groups and many other constructions in algebraic structures.

Quotient Groups

Quotient groups, denoted \( G/H \), arise when dividing a group \( G \) by one of its normal subgroups \( H \). The elements of a quotient group are the distinct cosets of \( H \) in \( G \). Each coset represents a distinct element of the new group. But how exactly do we "divide" such a complex structure like a group? Here's how it works:

• Each element in the quotient group corresponds to a set of elements from \( G \), specifically a coset of \( H \).
• The operation in the quotient group is defined by multiplying representatives of these cosets.
• This construction is only valid when \( H \) is normal in \( G \), ensuring that the operation does not depend on which representative of the coset is chosen.

Quotient groups are a powerful tool in algebra because they allow us to "simplify" a group by factoring out a normal subgroup, providing insights into the structure and properties of \( G \). They also play a critical role in studying the properties of group homomorphisms.

Algebraic Topology

Algebraic topology is the fascinating field that applies abstract algebra to topological spaces. Among its many powerful tools, the concept of group homomorphisms plays an essential role. By studying how groups map onto each other via homomorphisms, algebraic topologists can extract meaningful information about spaces.

• In algebraic topology, homomorphisms often arise from continuous functions between topological spaces, highlighting the intertwining of topology and algebra.
• The exploration of normal subgroups and quotient groups enables the simplification and study of complex continuous transformations.
• The consistent application in algebraic topology ensures a deepened understanding of how spaces can be deformed, divided, or connected.

Through these algebraic tools, algebraic topology creates a bridge between different areas of mathematics, revealing the deep, underlying structures of mathematical spaces. This connection helps us understand not just isolated geometric properties but how spaces are related and transformed under continuous mappings.