Fundamental Isomorphism Theorems

Why This Matters

The isomorphism theorems are the structural backbone of group theory—and by extension, everything you'll do in Galois Theory. When you're analyzing field extensions, you're constantly asking questions like "how does this quotient group relate to that subgroup?" or "what happens when I mod out by a normal subgroup?" These theorems give you the precise machinery to answer those questions. They transform messy, complicated group structures into clean isomorphisms you can actually work with.

You're being tested on your ability to apply these theorems, not just state them. That means recognizing when a problem is secretly asking you to use the First Isomorphism Theorem to identify a quotient, or when the Lattice Theorem explains why subgroups of a Galois group correspond to intermediate fields. Don't just memorize the formulas—know what structural insight each theorem provides and when to reach for it.

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Theorems That Build Quotients from Homomorphisms

The First Isomorphism Theorem is your workhorse. Whenever you have a homomorphism, you automatically get an isomorphism—the kernel tells you exactly what you're "collapsing" to create the quotient.

First Isomorphism Theorem

• Core statement—if $f: G \to H$ is a homomorphism, then $G/\ker(f) \cong \text{Im}(f)$
• The kernel determines everything: the kernel $\ker(f)$ is always normal in $G$, and the "size" of the quotient equals the "size" of the image
• Galois application: when you restrict automorphisms to a subfield, this theorem identifies the resulting quotient group with the automorphism group of the smaller extension

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Theorems That Relate Subgroups to Quotients

These theorems answer a natural question: if you have subgroups interacting inside a group, how do those relationships survive when you pass to quotients? The key insight is that normal subgroups act as "compatible" denominators.

Second Isomorphism Theorem

• Core statement—if $H \leq G$ and $N \trianglelefteq G$, then $HN/N \cong H/(H \cap N)$
• The "diamond" or "parallelogram" theorem: visualize $H$ and $N$ as two subgroups whose product $HN$ sits above them, with $H \cap N$ below
• Why it matters: lets you compute quotients of products without knowing $HN$ explicitly—you only need $H$ and the intersection

Third Isomorphism Theorem

• Core statement—if $K \subseteq N$ are both normal in $G$, then $(G/K)/(N/K) \cong G/N$
• "Cancellation" for quotients: modding out by $K$ first, then by $N/K$, gives the same result as modding out by $N$ directly
• Hierarchical structure: this theorem is essential when you're working with towers of field extensions and need to relate successive quotients

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Theorems That Preserve Lattice Structure

The Lattice Isomorphism Theorem (sometimes called the Fourth Isomorphism Theorem or Correspondence Theorem) reveals that passing to a quotient doesn't destroy subgroup structure—it preserves it in a precise, one-to-one way.

Fourth Isomorphism Theorem (Lattice Isomorphism Theorem)

• Core statement—there's a bijection between subgroups of $G$ containing $N$ and subgroups of $G/N$, given by $H \mapsto H/N$
• Lattice preservation: this bijection respects containment, intersections, and products—the "shape" of the subgroup lattice above $N$ matches the lattice of $G/N$
• Galois connection: this is why intermediate fields of $K/F$ correspond to subgroups of $\text{Gal}(K/F)$—the lattice structure transfers perfectly

Correspondence Theorem

• Generalized framework—extends the lattice correspondence to other algebraic structures (rings, modules) beyond groups
• Substructure preservation: normal subgroups above $N$ correspond to normal subgroups of the quotient, maintaining the "normal" property through the bijection
• Essential for Galois Theory: the Fundamental Theorem of Galois Theory is essentially this correspondence applied to automorphism groups and intermediate fields

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Quick Reference Table

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Self-Check Questions

1. If you have a surjective homomorphism $f: G \to H$, which theorem immediately tells you that $G/\ker(f) \cong H$? What must you verify about $\ker(f)$?

2. Compare the Second and Third Isomorphism Theorems: both produce isomorphisms involving quotients, but what's the key structural difference in their hypotheses?

3. You're given $K \trianglelefteq N \trianglelefteq G$. Write the isomorphism that the Third Isomorphism Theorem guarantees, and explain why this is useful for analyzing a tower of field extensions.

4. The Lattice Isomorphism Theorem says subgroups of $G/N$ correspond to subgroups of $G$ containing $N$. If $G$ has exactly 5 subgroups containing $N$, how many subgroups does $G/N$ have? What property do normal subgroups of $G$ (containing $N$) correspond to in $G/N$?

5. (FRQ-style) Let $f: G \to H$ be a homomorphism with $\ker(f) = N$. Suppose $K$ is a subgroup of $G$ containing $N$. Using the isomorphism theorems, describe $f(K)$ in terms of a quotient of $K$, and explain which theorem(s) you're applying.