Group theory

The deepest corner of abstract-algebra and the bulk of both texts here: roughly half of judson-abstract-algebra's chapters and six of lal-algebra-1's eleven. A group is the minimal structure in which an operation can be undone — associativity, an identity, inverses — and that minimum turns out to be exactly what symmetry requires.

Where groups come from

Judson's two founding examples are the ones the subject keeps returning to: integer equivalence classes under modular addition, and the symmetries of an equilateral triangle, whose six elements (three rotations, three reflections) compose non-commutatively. Lal names the same two sources from the other direction — "the two main sources of groups are the permutation groups and the matrix groups" — which is the same observation once you notice that both examples are groups of maps of a set to itself.

The results a first course is built around

• Lagrange's theorem — the order of a subgroup divides the order of the group. Judson derives Fermat's little theorem and Euler's theorem from it directly, which is where number theory re-enters as a consequence rather than a prerequisite.
• The isomorphism theorems, over normal subgroups and factor groups: the machinery for saying that two groups built differently are the same group, and for building new ones by quotient.
• The Sylow theorems — existence and conjugacy of subgroups of prime-power order, the main tool for classifying finite groups. Lal calls the classification of groups "the main problem in the theory."
• Classification of finite abelian groups — every one decomposes into cyclic pieces; both books reach it.
• Group actions and the class equation, with Burnside's counting theorem as the combinatorial payoff in Judson.

Lal goes further into structure theory than a first course usually does: Schreier and Jordan–Hölder, the Remak–Krull–Schmidt theorem on direct decomposition, solvable and nilpotent groups, and presentation theory. Judson spends that space on applications and on reaching Galois theory, where solvability of a group becomes solvability of an equation by radicals.

A third route in: arithmetic first (2026-10-04)

baker-algebra-number-theory puts groups after number-theory and uses them to repay it. Its chapter 2 covers permutations and the sign of a permutation, cycle type, symmetry groups, Lagrange's theorem, and group actions with the orbit-stabilizer theorem and Burnside's formula, applied to counting distinguishable colourings. The notable step is Theorem 2.28: a finite group with at most d solutions of x^d = ι for each divisor d of its order is cyclic, proved by comparing the number of elements of each order with the Euler function. Taking the units of ℤ/p gives the existence of primitive roots, which chapter 1 had stated without proof. Neither lal-algebra-1 nor judson-abstract-algebra proves it this way. Baker stops at Lagrange and actions: no Sylow, no isomorphism theorems.

Why it exports

The alternating group's simplicity, the solvable-group condition, and the matrix groups are each the group-theoretic half of something else — Galois theory, geometry, and the linear structures linear-algebra studies concretely. Symmetry arguments in physics and the modular arithmetic under public-key cryptography both run on the same axioms (judson-abstract-algebra chapter 7).

Related

abstract-algebra · number-theory · baker-algebra-number-theory · judson-abstract-algebra · lal-algebra-1 · linear-algebra · mathematics