Course Syllabus
This syllabus outlines the modules of Group Theory.
- CO1: The fundamental concepts of groups, subgroups, and cyclic structures.
- CO2: The properties of permutation groups, cosets, normal subgroups, and quotient groups.
- CO3: Group homomorphisms, the fundamental isomorphism theorems, and Cayley's theorem.
- CO4: Automorphisms, inner automorphisms, and commutator subgroups.
- CO5: Conjugacy classes, the class equation, Cauchy's theorem, and Sylow's theorems with applications.
- Binary operations, algebraic structures, and formal group axioms. Canonical examples of groups (numbers, modular arithmetic, matrices, Klein 4-group, dihedral groups, quaternion group $Q_8$). Elementary properties of groups (uniqueness of identity/inverses, reversal law, cancellation laws). Order of a group and order of an element. Subgroups and subgroup criteria (one-step, two-step, and finite subgroup tests). Subgroup intersections and unions. Center of a group and centralizer of an element. Generator of a group and cyclic groups. Properties and classification of cyclic groups (Fundamental Theorem of Cyclic Groups).
- Permutations and permutation groups: cycle notation, disjoint cycle decomposition, transpositions, and even/odd permutations. Alternating groups $A_n$ and their orders. Cosets of a subgroup: left and right cosets, properties of cosets, and coset partitions. Lagrange's Theorem and its immediate corollaries (order of elements, prime order groups). Euler's totient theorem and Fermat's Little Theorem as group-theoretic consequences. Normalizer of an element and normalizer of a subgroup. Normal subgroups: definition, equivalent criteria ($gHg^{-1} = H$, $aH = Ha$), and structural properties. Quotient groups (factor groups): well-definedness of coset multiplication and construction of $G/H$. Center of a group as a normal subgroup, and the $G/Z(G)$ cyclic $\implies G$ abelian theorem. Subgroups of index 2, simple groups definition, and non-simplicity criteria.
- Group homomorphisms: definition, elementary properties, preservation of identity, inverses, and subgroups. Kernel and image of a homomorphism, and the normality of the kernel ($\ker \phi \trianglelefteq G$). Trivial kernel criterion for injectivity ($\ker \phi = \{e\}$). Group isomorphisms: definition, isomorphism as an equivalence relation, and isomorphic classification. First Isomorphism Theorem (Fundamental Theorem of Homomorphisms: $G/\ker \phi \cong \operatorname{Im} \phi$). Applications of the First Isomorphism Theorem to quotient groups. Second Isomorphism Theorem (Diamond / Butterfly Theorem: $HN/N \cong H/(H \cap N)$). Third Isomorphism Theorem (Quotient of Quotients Theorem: $(G/K)/(N/K) \cong G/N$ for $K \trianglelefteq G, K \subseteq N \trianglelefteq G$). Fourth Isomorphism Theorem (Correspondence / Lattice Theorem for subgroups). Cayley's Theorem: representation of every group as a permutation group ($\operatorname{Sym}(G)$). Left regular representation and permutation embeddings.
- Automorphisms: definition, automorphism group $\operatorname{Aut}(G)$, and group structure of automorphisms. Inner automorphisms: conjugation mappings $\phi_g(x) = gxg^{-1}$ and the inner automorphism group $\operatorname{Inn}(G)$. Normality of $\operatorname{Inn}(G)$ in $\operatorname{Aut}(G)$ and the canonical isomorphism $\operatorname{Inn}(G) \cong G/Z(G)$. Outer automorphisms and the outer automorphism group $\operatorname{Out}(G) = \operatorname{Aut}(G)/\operatorname{Inn}(G)$. Automorphism groups of cyclic groups: $\operatorname{Aut}(\mathbb{Z}_n) \cong \mathbb{U}(n)$ and $\operatorname{Aut}(\mathbb{Z}) \cong \mathbb{Z}_2$. Characteristic subgroups: definition, distinction from normal subgroups, and transitivity of characteristic subgroups. Commutator of two elements and the commutator subgroup (derived subgroup) $G'$. Normality of $G'$ in $G$, abelianness of the factor group $G/G'$ (abelianization), and the universal property of abelianization.
- Conjugacy relation on a group: definition, equivalence relation, and conjugacy classes. Conjugacy class size formula: $|\operatorname{cl}(a)| = [G : C_G(a)]$ (orbit-stabilizer connection). The Class Equation: $|G| = |Z(G)| + \sum [G : C_G(a_i)]$ and its arithmetic structure. Conjugacy classes in symmetric groups $S_n$ via cycle type. $p$-groups: definition and prime-power order structures. Non-triviality of the center of a non-trivial finite $p$-group ($Z(G) \ne \{e\}$). Classification of groups of order $p^2$ (every group of order $p^2$ is abelian: $\mathbb{Z}_{p^2}$ or $\mathbb{Z}_p \times \mathbb{Z}_p$). Cauchy's Theorem for finite abelian and non-abelian groups. Sylow $p$-subgroups: definition and prime-power divisor structure. First Sylow Theorem: existence of Sylow $p$-subgroups. Second Sylow Theorem: conjugacy of Sylow $p$-subgroups. Third Sylow Theorem: counting formula $n_p \equiv 1 \pmod p$ and $n_p \mid m$. Sylow normality and uniqueness criterion ($n_p = 1 \iff P \trianglelefteq G$). Applications of Sylow Theorems: non-simplicity tests and classification of groups of order $pq$.
- Gallian, Joseph A. (2021). Contemporary Abstract Algebra (10th ed.). Chapman and Hall/CRC.
- Dummit, David S., & Foote, Richard M. (2004). Abstract Algebra (3rd ed.). John Wiley & Sons.
- Herstein, I. N. (1975). Topics in Algebra (2nd ed.). John Wiley & Sons.
- Khanna, V. K., & Bhambri, S. K. (2017). A Course in Abstract Algebra (5th ed.). Vikas Publishing House.
- Vashistha, A. R. Modern Algebra (Group Theory). Krishna Prakashan Media.