Summary of Key Results

The following table provides a quick reference for the major theorems and their locations in this document.

The key dependencies among topics are:

• Groups $\to$ Subgroups $\to$ Lagrange’s Theorem $\to$ Normal Subgroups $\to$ Quotient Groups
• Homomorphisms $\to$ Isomorphism Theorems $\to$ Correspondence Theorem
• Group Actions $\to$ Orbit-Stabilizer $\to$ Sylow Theorems $\to$ Applications
• Rings $\to$ Ideals $\to$ Quotient Rings $\to$ Prime/Maximal Ideals
• Polynomial Rings $\to$ Euclidean Domains $\to$ PIDs $\to$ UFDs
• Field Extensions $\to$ Algebraic Extensions $\to$ Splitting Fields $\to$ Galois Theory

Mastering the earlier topics is essential before proceeding to the later ones. The problem set (Section 18) Is designed to test understanding across all these areas.

• Confusing groups, rings, and fields. Group: one operation, inverses, identity. Ring: two operations (addition, multiplication). Field: ring where every non-zero element has a multiplicative inverse. Fix: $\mathbb{Z}$ is a ring but not a field ($2$ has no multiplicative inverse). $\mathbb{Q}$, $\mathbb{R}$, $\mathbb{C}$ are fields.
• Wrong subgroup test. A subset $H \leq G$ must contain the identity, be closed under the operation and inverses. Fix: For finite groups, it suffices to check closure: if $H$ is non-empty and closed, then $H \leq G$.
• Confusing normal subgroups and general subgroups. Normal subgroups satisfy $gHg^{-1} = H$ for all $g \in G$; they allow quotient group construction. Fix: Abelian groups: all subgroups are normal. Non-abelian: check $gH = Hg$ for all $g$.

Problem. Show that $(\mathbb{Z}_n, +)$ is a group for any $n \in \mathbb{N}$.

Solution. (1) Closure: $[a] + [b] = [a + b \bmod n] \in \mathbb{Z}_n$. (2) Associativity: inherited from $\mathbb{Z}$. (3) Identity: $[0]$. (4) Inverse: $[a]^{-1} = [-a] = [n - a]$. Therefore $(\mathbb{Z}_n, +)$ is a group. $\blacksquare$

Problem. $G = S_3$ has order 6. A subgroup $H = \{e, (12)\}$ has order 2. Verify Lagrange’s theorem.

Solution. $|G| = 6$, $|H| = 2$. $2 \mid 6$, so Lagrange’s theorem is satisfied: $|H|$ divides $|G|$. The index $[G : H] = |G|/|H| = 3$, and the cosets are $H$, $(13)H = \{(13), (132)\}$, $(23)H = \{(23), (123)\}$.

$\blacksquare$

• Group: set with associative binary operation, identity, inverses. Abelian if commutative.
• Lagrange’s theorem: $|H|$ divides $|G|$ for $H \leq G$; the index $[G:H] = |G|/|H|$.
• Cyclic group: generated by a single element $\langle g \rangle$; $\mathbb{Z}_n$ is cyclic.
• Ring: abelian group under $+$, monoid under $\times$, distributive laws. Field: ring where every non-zero element is invertible.

• Extension degree and group order. For a Galois extension $E/F$: $[E:F]=|\mathrm{Gal}(E/F)|$. This links field theory to group theory directly.
• Normal subgroups and quotient groups. $K/F$ is Galois $\iff$ $\mathrm{Gal}(E/K)\trianglelefteq\mathrm{Gal}(E/F)$, and in that case $\mathrm{Gal}(K/F)\cong\mathrm{Gal}(E/F)/\mathrm{Gal}(E/K)$.