Group theory

Mathematics (B.Sc) · Algebra · free study notes for exams, placements and interviews

In short

A group is a set with one associative binary operation that has an identity and an inverse for every element - the minimum structure needed to talk about symmetry and reversible operations. Group theory is the first abstract algebra course in every B.Sc Mathematics CBCS syllabus and the foundation for rings, fields, vector spaces and Galois theory. It is examined heavily in university papers and in IIT JAM, GATE (MA), CSIR-UGC NET and state SET papers, where Lagrange's theorem, permutation groups, quotient groups and Sylow's theorems carry the bulk of the marks. Industrially it underlies public-key cryptography (RSA, Diffie-Hellman, elliptic-curve groups), error-correcting and check-digit codes - the Aadhaar number's 12th digit is a Verhoeff checksum built on the dihedral group of order 10 - and crystal and molecular symmetry classification.

Explained

Groups, subgroups and first consequences

A binary operation on a set G is a map G × G → G (closure is part of the definition).
Group (G, ∗): (i) closure, (ii) associativity a∗(b∗c) = (a∗b)∗c, (iii) identity e with a∗e = e∗a = a, (iv) inverse a⁻¹ with a∗a⁻¹ = e. Abelian if a∗b = b∗a for all a, b.
Weaker structures: semigroup (closure + associativity), monoid (semigroup + identity), group (monoid + inverses). A quasigroup/loop drops associativity - not examined much.
Uniqueness results: identity is unique; each inverse is unique; (a⁻¹)⁻¹ = a; (ab)⁻¹ = b⁻¹a⁻¹ (note the reversal); cancellation laws ab = ac ⇒ b = c hold in every group.
Standard examples: (ℤ, +), (ℚ∗, ×), (ℝⁿ, +), ℤₙ under addition mod n, U(n) = units mod n under multiplication (order φ(n)), GLₙ(ℝ) and SLₙ(ℝ) of matrices, Sₙ (permutations), Aₙ (even permutations), Dₙ (symmetries of a regular n-gon, order 2n), the quaternion group Q₈, Klein four-group V₄, groups of nth roots of unity, and the circle group.
Subgroup H ≤ G: a subset that is itself a group under the same operation. One-step test: H ≠ ∅ and a, b ∈ H ⇒ ab⁻¹ ∈ H. For a *finite* subset, closure alone suffices.
Centre Z(G) = {z : zg = gz ∀g} is a subgroup; centraliser C(a) = {g : ga = ag}; normaliser N(H) = {g : gHg⁻¹ = H}. All are subgroups.
Intersection of subgroups is a subgroup; a union generally is not (2ℤ ∪ 3ℤ is not closed).
Order of a group |G| = number of elements; order of an element o(a) = least n > 0 with aⁿ = e (infinite if none exists).

Cyclic groups

G is cyclic if G = ⟨a⟩ = {aⁿ : n ∈ ℤ} for some generator a. Every cyclic group is abelian; the converse is false (V₄ is abelian, not cyclic).
Up to isomorphism there are only two kinds: ℤ (infinite) and ℤₙ (order n).
Every subgroup of a cyclic group is cyclic. In ℤₙ there is exactly one subgroup of order d for each divisor d of n, namely ⟨n/d⟩ - so the number of subgroups equals the number of divisors of n.
Generators of ℤₙ are the residues coprime to n, hence φ(n) of them. In a cyclic group of order n, the number of elements of order d (d | n) is φ(d), and Σ_{d|n} φ(d) = n.
o(aᵏ) = n / gcd(n, k) where o(a) = n. aᵏ = e ⇔ n | k.
Every group of prime order p is cyclic, hence ≅ ℤₚ, hence abelian and simple.
ℤₘ × ℤₙ ≅ ℤ_{mn} ⇔ gcd(m, n) = 1. So ℤ₂ × ℤ₃ ≅ ℤ₆ but ℤ₂ × ℤ₂ ≇ ℤ₄.

Permutation groups

A permutation of a finite set is a bijection onto itself; Sₙ is the group of all of them under composition, |Sₙ| = n!. Sₙ is non-abelian for n ≥ 3.
Every permutation is a product of disjoint cycles (unique up to order); disjoint cycles commute. A k-cycle has order k, and o(σ) = lcm of its cycle lengths.
A k-cycle = product of k − 1 transpositions. Parity (even/odd) is well defined; sgn σ = (−1)^{number of transpositions}. Even permutations form Aₙ, with |Aₙ| = n!/2 for n ≥ 2.
Cycle type = partition of n; two permutations in Sₙ are conjugate ⇔ same cycle type. Conjugation relabels: τστ⁻¹ is σ with each entry replaced by its τ-image.
Sₙ is generated by transpositions, also by (1 2) and (1 2 … n); Aₙ is generated by 3-cycles.
Cayley's theorem: every group of order n embeds in Sₙ (via left translation g ↦ Lg) - so every finite group is a permutation group.
Aₙ is simple for n ≥ 5; A₄ is not (V₄ ⊲ A₄). This is the algebraic reason the general quintic is not solvable by radicals.
Dihedral group Dₙ = ⟨r, s | rⁿ = s² = e, srs = r⁻¹⟩, order 2n - n rotations and n reflections. D₅ (order 10) drives the Verhoeff check-digit scheme used in Aadhaar.

Cosets, Lagrange, normal subgroups and quotient groups

Left coset aH = {ah : h ∈ H}; right coset Ha. Cosets of H partition G and all have |H| elements; aH = bH ⇔ b⁻¹a ∈ H.
Lagrange's theorem: for finite G and H ≤ G, |G| = |H| · [G : H], so |H| divides |G|. Corollaries: o(a) divides |G|; a^|G| = e; a group of prime order is cyclic.
Converse of Lagrange is false: A₄ has order 12 but no subgroup of order 6. Partial converses: Cauchy's theorem (if p | |G|, G has an element - hence a subgroup - of order p) and Sylow's first theorem for prime powers.
Number-theoretic payoff: Euler's theorem a^φ(n) ≡ 1 (mod n) for gcd(a, n) = 1, and Fermat's little theorem a^p−1 ≡ 1 (mod p) - both are Lagrange applied to U(n). These are the arithmetic engine of RSA.
Normal subgroup N ⊴ G: gNg⁻¹ = N for all g, equivalently gN = Ng, equivalently every left coset is a right coset. Any subgroup of index 2 is normal; Z(G) and kernels are always normal; every subgroup of an abelian group is normal.
Quotient (factor) group G/N: cosets with (aN)(bN) = abN - well defined *only* when N is normal. |G/N| = [G : N].
G/Z(G) cyclic ⇒ G abelian. Simple group: no normal subgroups except {e} and G.
Correspondence: subgroups of G/N ↔ subgroups of G containing N.

Homomorphisms and the isomorphism theorems

Homomorphism f : G → H with f(ab) = f(a)f(b). Then f(e_G) = e_H, f(a⁻¹) = f(a)⁻¹, and o(f(a)) divides o(a).
Kernel ker f = {g : f(g) = e_H} ⊴ G; image f(G) ≤ H. f is injective ⇔ ker f = {e}. Isomorphism = bijective homomorphism; automorphism = isomorphism G → G, and Aut(G) is a group; inner automorphisms Inn(G) ≅ G/Z(G).
First isomorphism theorem: G/ker f ≅ f(G). (Example: det : GLₙ(ℝ) → ℝ∗ has kernel SLₙ(ℝ), so GLₙ(ℝ)/SLₙ(ℝ) ≅ ℝ∗.)
Second (diamond): for H ≤ G, N ⊴ G: HN/N ≅ H/(H ∩ N). Third: for N ⊆ M, both normal, (G/N)/(M/N) ≅ G/M.
Isomorphism invariants used to prove two groups are *not* isomorphic: order, abelian or not, number of elements of each order, cyclic or not, centre, subgroup lattice.
Aut(ℤₙ) ≅ U(n), of order φ(n).

Group actions, class equation and Sylow's theorems

A (left) action of G on a set X: g·x with e·x = x and (gh)·x = g·(h·x); equivalently a homomorphism G → Sym(X). Orbit O(x) = {g·x}; stabiliser Gₓ = {g : g·x = x} ≤ G.
Orbit-stabiliser theorem: |O(x)| = [G : Gₓ] = |G|/|Gₓ|; orbits partition X.
Conjugacy is the action g·x = gxg⁻¹. Conjugacy classes partition G; |cl(a)| = [G : C(a)]; a ∈ Z(G) ⇔ |cl(a)| = 1.
Class equation: |G| = |Z(G)| + Σ [G : C(aᵢ)] over one representative of each class of size > 1.
Consequences for p-groups (|G| = pⁿ): Z(G) ≠ {e}; every group of order p² is abelian (≅ ℤ_{p²} or ℤₚ × ℤₚ); a p-group has a normal subgroup of each order p^k.
Burnside's (Cauchy-Frobenius) lemma: number of orbits = (1/|G|) Σ_{g∈G} |Fix(g)| - the counting tool for necklaces, colourings and dice.
Sylow's theorems (|G| = pᵏm, p ∤ m): (1) a subgroup of order pᵏ (Sylow p-subgroup) exists; (2) all Sylow p-subgroups are conjugate; (3) their number nₚ divides m and nₚ ≡ 1 (mod p). nₚ = 1 ⇔ that Sylow subgroup is normal.
Typical use: prove a group of a given order is not simple, or is cyclic, or must be abelian. Groups of order pq (p < q, p ∤ q − 1) are cyclic.
Structure theorem for finite abelian groups: every finite abelian group is a direct product of cyclic groups of prime-power order, uniquely up to ordering - so the abelian groups of order 8 are ℤ₈, ℤ₄ × ℤ₂, ℤ₂ × ℤ₂ × ℤ₂ (three in all, matching the partitions of 3).
Series: normal series, composition series (Jordan-Hölder uniqueness), solvable groups (abelian factors); Sₙ is solvable only for n ≤ 4.

Key points

Group - set with an associative operation, identity and inverses; abelian if the operation commutes.
Subgroup test - non-empty and closed under ab⁻¹; for finite subsets, closure alone is enough.
Order of an element - least n with aⁿ = e; it always divides |G| in a finite group.
Cyclic group - generated by one element; ℤ or ℤₙ; all subgroups cyclic, one per divisor.
φ(n) - number of generators of ℤₙ and the order of U(n); elements of order d number φ(d).
Lagrange's theorem - |H| divides |G|; its converse is false (A₄ has no subgroup of order 6).
Cauchy's theorem - p | |G| guarantees an element of order p.
Normal subgroup - gNg⁻¹ = N; needed for the quotient G/N to be a group; index 2 ⇒ normal.
Kernel - always normal, and G/ker f ≅ im f (first isomorphism theorem).
Sₙ, Aₙ - orders n! and n!/2; order of a permutation = lcm of cycle lengths; conjugate ⇔ same cycle type.
Cayley's theorem - every group of order n sits inside Sₙ.
Dihedral Dₙ - order 2n, rⁿ = s² = e, srs = r⁻¹; the standard non-abelian example.
Orbit-stabiliser - |orbit| × |stabiliser| = |G|; conjugacy class size = index of the centraliser.
Class equation - |G| = |Z(G)| + Σ indices of centralisers; forces Z(G) ≠ {e} in a p-group.
Sylow - existence, conjugacy, and nₚ | m with nₚ ≡ 1 mod p; nₚ = 1 means normal, so G is not simple.
Finite abelian structure - unique product of prime-power cyclic groups; count by partitions of exponents.
Simple groups - ℤₚ and Aₙ (n ≥ 5); A₅, order 60, is the smallest non-abelian simple group.
Cryptographic link - Euler/Fermat (Lagrange in U(n)) underpin RSA; discrete logs live in cyclic groups.

Formulas and examples

|G| = |H| × [G : H] (Lagrange; |H| = order of subgroup, [G : H] = number of distinct cosets)
o(aᵏ) = n / gcd(n, k), where o(a) = n; a^|G| = e for finite G
|ℤₙ| = n; generators = φ(n); number of subgroups = d(n) (number of divisors); elements of order d = φ(d)
|U(n)| = φ(n); a^φ(n) ≡ 1 (mod n) for gcd(a, n) = 1; a^p−1 ≡ 1 (mod p)
|Sₙ| = n!, |Aₙ| = n!/2, |Dₙ| = 2n, |GL₂(𝔽ₚ)| = (p² − 1)(p² − p)
o(σ) = lcm(cycle lengths); sgn σ = (−1)^(n − number of disjoint cycles including fixed points)
|orbit of x| = [G : Gₓ] = |G| / |stabiliser|; |cl(a)| = |G| / |C(a)|
|G| = |Z(G)| + Σᵢ [G : C(aᵢ)] (class equation)
Number of orbits = (1/|G|) Σ_{g∈G} |Fix(g)| (Burnside)
|G| = pᵏ·m, p ∤ m: |Sylow p-subgroup| = pᵏ, nₚ | m, nₚ ≡ 1 (mod p)
|HK| = |H||K| / |H ∩ K|; G/ker f ≅ im f

Example 1 - orders and subgroups of a cyclic group. G = ℤ₆₀ = ⟨1⟩.

Divisors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 → 12 subgroups, one of each of those orders.
Generators: φ(60) = 60 × (1 − 1/2)(1 − 1/3)(1 − 1/5) = 60 × 1/2 × 2/3 × 4/5 = 16.
o(18) = 60/gcd(60, 18) = 60/6 = 10. Elements of order 10: φ(10) = 4.
Check: Σ_{d|60} φ(d) = 1+1+2+2+4+2+4+4+8+8+8+16 = 60 ✓.

Example 2 - a permutation fully analysed. σ ∈ S₈ with 1→3, 2→5, 3→4, 4→1, 5→2, 6→7, 7→6, 8→8.

Disjoint cycles: σ = (1 3 4)(2 5)(6 7), with 8 fixed.
o(σ) = lcm(3, 2, 2) = 6.
Transpositions: (1 3 4) = (1 4)(1 3) → 2; each 2-cycle → 1. Total 4 → even, sgn σ = +1, so σ ∈ A₈.
σ⁻¹ = (1 4 3)(2 5)(6 7). Conjugating by τ = (1 2): τστ⁻¹ = (2 3 4)(1 5)(6 7) - same cycle type, as required.

Example 3 - structure of U(15). U(15) = {1, 2, 4, 7, 8, 11, 13, 14}, order φ(15) = 8.

Orders: 2 → 2, 4, 8, 16 ≡ 1, so o(2) = 4; 4² = 16 ≡ 1 → o(4) = 2; 14 ≡ −1 → o(14) = 2; 11² = 121 = 120 + 1 ≡ 1 → o(11) = 2.
No element of order 8 (max is 4), so U(15) is not cyclic; with ⟨2⟩ ≅ ℤ₄ and ⟨11⟩ ≅ ℤ₂ intersecting trivially, U(15) ≅ ℤ₄ × ℤ₂.
Euler check: 2⁸ = 256 = 17 × 15 + 1 ≡ 1 (mod 15) ✓.

Example 4 - Sylow: every group of order 15 is cyclic. |G| = 15 = 3 × 5.

n₃ | 5 and n₃ ≡ 1 (mod 3) → n₃ ∈ {1, 5}; 5 ≢ 1 (mod 3), so n₃ = 1. n₅ | 3 and n₅ ≡ 1 (mod 5) → n₅ = 1.
So P₃ ≅ ℤ₃ and P₅ ≅ ℤ₅ are both normal, P₃ ∩ P₅ = {e} (orders coprime), |P₃P₅| = 15/1 = 15 = |G|.
Hence G ≅ ℤ₃ × ℤ₅ ≅ ℤ₁₅, cyclic. (Same argument: order 20 has n₅ = 1, so no group of order 20 is simple.)

Example 5 - class equation of D₄ and Burnside counting. D₄ = {e, r, r², r³, s, rs, r²s, r³s}, |D₄| = 8.

Classes: {e}, {r²}, {r, r³}, {s, r²s}, {rs, r³s} → 1 + 1 + 2 + 2 + 2 = 8 ✓, so Z(D₄) = {e, r²} and |cl(r)| = 2 = 8/|C(r)| → |C(r)| = 4.
Burnside: colour the 4 corners of a square with 2 colours, rotations only (G = ℤ₄). Fix(e) = 2⁴ = 16, Fix(r) = 2, Fix(r²) = 2² = 4, Fix(r³) = 2. Orbits = (16 + 2 + 4 + 2)/4 = 24/4 = 6 distinct colourings.

Group	Order	Abelian?	Cyclic?	Note
ℤₙ	n	Yes	Yes	one subgroup per divisor
ℤ₂ × ℤ₂ (V₄)	4	Yes	No	every non-identity element has order 2
S₃ ≅ D₃	6	No	No	smallest non-abelian group
D₄	8	No	No	Z = {e, r²}; 5 conjugacy classes
Q₈	8	No	No	one element of order 2; every subgroup normal
A₄	12	No	No	no subgroup of order 6
A₅	60	No	No	smallest non-abelian simple group

Where it is used

Cryptography and digital payments: RSA (U(n) and Euler's theorem), Diffie-Hellman and ECC (cyclic groups of prime order) secure UPI, Aadhaar eKYC, digital signatures under the Information Technology Act, 2000 and CCA-licensed certifying authorities.
Check digits and error control: the Verhoeff algorithm on D₅ gives the Aadhaar 12th digit; cyclic and BCH/Reed-Solomon codes in telecom and storage are built on group and field structure.
Chemistry, physics and materials: point groups and the 230 crystallographic space groups classify crystals and molecular vibrations - used in XRD work at CSIR labs, IITs and pharma R&D.
Robotics, graphics and CAD: rigid-body motions form SO(3) and SE(3); rotation composition and inverse kinematics are group operations.
Job roles and recruiters: cryptography and security engineers (C-DAC, ISRO, DRDO, TCS/Wipro security practices), quant and data-science roles that screen on discrete mathematics, and teaching/research posts filled through CSIR-UGC NET and GATE (MA).
Competitive examination load: IIT JAM MA, GATE MA, TIFR/ISI entrance and NET papers all test Lagrange, Sylow, permutation groups and the abelian structure theorem almost every year.

Interview questions

Q: What four axioms define a group? A: Closure under an associative binary operation, existence of an identity element, and an inverse for every element; commutativity is extra and makes it abelian.
Q: Is every abelian group cyclic? A: No - the Klein four-group ℤ₂ × ℤ₂ is abelian but has no element of order 4, so no single generator; every cyclic group is abelian, though.
Q: State Lagrange's theorem and whether its converse holds. A: The order of a subgroup divides the order of a finite group; the converse fails - A₄ has order 12 but no subgroup of order 6.
Q: Why must N be normal to form G/N? A: The product (aN)(bN) = abN is well defined only if aN = Na for all a; otherwise different coset representatives give different products.
Q: How do you find the order of a permutation? A: Write it as disjoint cycles and take the lcm of the cycle lengths - for (1 3 4)(2 5)(6 7) it is lcm(3,2,2) = 6.
Q: When are two permutations in Sₙ conjugate? A: Exactly when they have the same cycle type (same multiset of cycle lengths).
Q: What does Cayley's theorem say? A: Every group of order n is isomorphic to a subgroup of Sₙ, via the left-translation maps - so abstract finite groups are concrete permutation groups.
Q: Prove any group of prime order is cyclic. A: Take a ≠ e; o(a) divides p and is not 1, so o(a) = p and ⟨a⟩ = G.
Q: Why is the centre of a p-group non-trivial? A: In the class equation every non-central class has size a positive power of p, so |Z(G)| ≡ |G| ≡ 0 mod p, forcing |Z(G)| ≥ p.
Q: Use Sylow to show no group of order 20 is simple. A: n₅ divides 4 and is ≡ 1 mod 5, so n₅ = 1; that unique Sylow 5-subgroup is normal and proper, so G is not simple.
Q: How many abelian groups of order 8 are there, and why? A: Three - ℤ₈, ℤ₄ × ℤ₂, ℤ₂³ - one for each partition of the exponent 3, by the structure theorem.
Q: Where does group theory touch real software? A: Modular-arithmetic groups give RSA and Diffie-Hellman key exchange, elliptic-curve groups give compact signatures, and dihedral-group check digits validate Aadhaar numbers.

Common mistakes

Treating the converse of Lagrange as true, or assuming a divisor of |G| must be an element's order (Cauchy guarantees it only for primes).
Forgetting closure or associativity when verifying a group, and assuming a union of subgroups is a subgroup.
Writing (ab)⁻¹ = a⁻¹b⁻¹ instead of b⁻¹a⁻¹ in a non-abelian group.
Forming G/H when H is not normal, or confusing "H normal" with "H abelian" or with "every element of H commutes with every element of G".
Composing permutations in the wrong order (στ usually means "τ first"), and losing fixed points when counting parity.
Confusing order of an element with order of the group, or claiming ℤₘ × ℤₙ ≅ ℤ_{mn} without checking gcd(m, n) = 1.
Mis-stating Sylow's third theorem - nₚ must divide the *co-factor* m and satisfy nₚ ≡ 1 mod p; checking only one condition gives wrong counts.