Groups: Definitions, Theorems, Subgroups, and Cyclic Groups (Comprehensive Notes)

Binary Operation and Groups

• Binary operation on a set G (often called multiplication) is a function that assigns to each ordered pair (a,b) ∈ G×G an element in G. This condition is called closure: ab ∈ G for all a,b ∈ G.

• The most familiar binary operations are addition, subtraction, and multiplication on the integers. Some operations are not closed (e.g., division on the integers gives non-integers).

• Addition modulo n and multiplication modulo n on the set Z_n = {0,1,2,…,n−1} are important examples in abstract algebra; context determines whether we use addition modulo n, multiplication modulo n, or both.

Definition: Group

• A group is a set G together with a binary operation (usually called multiplication) that assigns to each ordered pair (a,b) ∈ G×G an element ab ∈ G, satisfying three properties:

  1. Associativity: (ab)c = a(bc) for all a,b,c ∈ G.

  2. Identity: There exists an element e ∈ G such that ae = ea = a for all a ∈ G.

  3. Inverses: For each a ∈ G there exists b ∈ G (an inverse of a) such that ab = ba = e.

• In words: a group is a set with an associative operation, an identity, inverses for every element, and closure (elements stay in the set after the operation).

• If ab = ba for all a,b ∈ G, the group is Abelian (commutative). If not, the group is non-Abelian.

• Important note: if a is the inverse of b, then b is the inverse of a.

• When testing a substance for being a group, always verify closure (Example 5 in the text).

Notation and Conventions

• A group can be represented multiplicatively as (G,·) with product ab; or additively with + and e denoting the identity (often written 0). In additive notation, inverses are written as −a and exponents as multiples, e.g., na.

• The notation g^n (n ∈ Z) denotes the n-fold product of g with itself: g^1 = g, g^2 = gg, g^0 = e, g^{−n} = (g^{−1})^n, etc.

• For abelian groups, the laws of exponents simplify similarly to integers, but in general groups (especially non-Abelian) we have (ab)^n ≠ a^n b^n.

• The inverse of a product is (ab)^{-1} = b^{-1} a^{-1} (Socks–Shoes property).

Theorems about Existence and Uniqueness

• Theorem (Uniqueness of the identity): In a group, there is a unique identity element.

• Theorem (Cancellation): In a group, right and left cancellation hold; if ba = ca or ab = ac, then b = c.

• Theorem (Uniqueness of inverses): Each element a in a group has a unique inverse a^{-1} with aa^{-1} = a^{-1}a = e.

• Consequence: In a Cayley table, each group element appears exactly once in each row and each column (a consequence of cancellation).

• Theorem (Socks–Shoes): For a,b in G, (ab)^{-1} = b^{-1} a^{-1}.

Example Highlights (Illustrative Groups)

• Example 1: Z, Q, R under ordinary addition are groups with identity 0 and inverses −a.

• Example 2: The set of integers under ordinary multiplication is not a group since 1 is the identity but not every element has a multiplicative inverse (e.g., 5 has no integer b with 5b = 1).

• Example 3: The subset {1, −1, i, −i} of C is a group under complex multiplication; −1 is its own inverse; i’s inverse is −i, etc.

• Example 4: The set Q_{>0} of positive rationals is a group under ordinary multiplication; inverse of a is 1/a.

• Example 5: The set of positive irrational numbers together with 1 under multiplication satisfies closure and associativity and has an identity, but is not a group because it is not closed (e.g., 2⋅2 = 4 ∉ the set if 2 is rational, illustrating lack of closure).

• Example 6: The set of all 2×3 matrices with real entries is a group under componentwise addition; the identity is the zero matrix; the inverse is the negative of the matrix.

• Example 7: The set Z_n under addition modulo n is a group; identity 0; inverse of k is n−k.

• Example 8: The set R^* of nonzero real numbers is a group under ordinary multiplication; identity 1; inverse 1/x.

• Example 9: The set GL(2, R) of 2×2 matrices with nonzero determinant is a non-Abelian group under matrix multiplication. det(AB) = (det A)(det B) ensures closure of nonzero determinants; the identity is the 2×2 identity matrix; inverses exist and have a closed form via adjugate and determinant.

• Example 10: The set of all 2×2 real matrices is not a group under multiplication because not all matrices are invertible (det = 0).

• Example 11: U(n) = {positive integers < n that are relatively prime to n} forms a group under multiplication modulo n; example U(10) = {1,3,7,9} with table shown; for prime n, U(n) = {1,…,n−1} is cyclic of order n−1.

• Example 12: The set {0,1,2,3} mod 4 is not a group because 0 and 2 do not have inverses.

• Example 13: The set of integers under subtraction is not a group (not associative).

• Example 14: The complex numbers C under addition form a group; C^* (nonzero complex numbers) under multiplication forms a non-Abelian? Actually C^* is Abelian under multiplication; inverse of a+bi is (a−bi)/(a^2+b^2).

• Example 15: The set of all complex nth roots of unity is a group under multiplication; these are the zeros of x^n − 1 in C, with geometric interpretation on the unit circle; De Moivre’s theorem connects to their arguments.

• Example 16: R^n with componentwise addition is a group.

• Example 17-18: GL(2, F) for F ∈ {Q, R, C} or F = Z_p (p prime) are non-Abelian in general; SL(2, F) (determinant 1) is another family of non-Abelian groups under multiplication.

• Example 19: The set {1,2,…,n^2−1} under multiplication modulo n is a group if and only if n is prime (this aligns with unit group properties modulo n).

Table: Summary of Common Group Examples

• Form: Group, Operation, Identity, Inverse, Abelian?

• Z under addition, abelian, inverse of a is −a.

• Q_1 under multiplication, abelian, inverse of a is 1/a.

• Z_n under addition (mod n), abelian, inverse of k is n−k.

• R^* under multiplication, abelian, inverse 1/x.

• C^* under multiplication, abelian, inverse ā/(|a|^2) for a ≠ 0.

• GL(2, F) (F any field or Z_p), non-Abelian in general, defined by determinant ≠ 0, matrix multiplication.

• U(n) under multiplication mod n, abelian for some n, often nontrivial structure (cyclic for prime n).

• D_n (dihedral group) and other matrix groups: often non-Abelian; generator relations vary.

The Goal of Abstract Algebra

• The aim is to discover truths about algebraic systems (sets with one or more binary operations) that depend only on the axioms (properties) of those operations, not on the specific nature of the elements.

• This abstraction allows general deductions that apply to many concrete groups.

Finite Groups; Subgroups

• The order of a group |G| is the number of elements in G; the order of an element g ∈ G is the smallest positive integer n with g^n = e, if such n exists; otherwise g has infinite order. Notation: |G|, |g|.

• Example: |U(15)| = 8; orders of elements can be computed by raising elements to powers until identity.

• If a has order n, it generates the cyclic subgroup ⟨a⟩ = {a^k | k ∈ Z} of order n. If n is infinite, ⟨a⟩ ≈ Z; if n is finite, ⟨a⟩ = {e, a, a^2, …, a^{n−1}} and is cyclic.

• Cyclic subgroups are always abelian (because ⟨a⟩ is generated by a single element, so any two elements commute).

• The cyclic group generated by a is denoted ⟨a⟩; the subgroup generated by a single element is cyclic.

• Theorem (Criterion for ai = aj in ⟨a⟩): If |a| = ∞, then a^i = a^j iff i = j; if |a| = n, then ⟨a⟩ = {e, a, …, a^{n−1}} and a^i = a^j iff n divides i−j.

• Corollaries: (i) |a| ≤ |⟨a⟩|; (ii) a^k = e implies |a| divides k; (iii) If a,b ∈ finite G and ab = ba, then |ab| divides |a||b|.

• The order of a^k in a group with a of order n is |a^k| = n / gcd(n,k).

• Theorem 4.2 (order of powers): If |a| = n and k > 0, then |a^k| = n / gcd(n,k). Also, |⟨a^k⟩| = n / gcd(n,k).

• Corollaries: If gcd(n,k) = 1, then ⟨a^k⟩ = ⟨a⟩ and a^k is a generator of ⟨a⟩. In particular, for Zn, k is a generator iff gcd(n,k) = 1.

Generators and Subgroups of Cyclic Groups

• Corollary 3: If |a| = n, then ⟨a^j⟩ is a subgroup of ⟨a⟩ iff gcd(n,j) > 0; |a^j| = n / gcd(n,j). In particular, ⟨a^j⟩ is all of ⟨a⟩ iff gcd(n,j) = 1.

• Corollary 4: In Zn, k is a generator (i.e., ⟨k⟩ = Zn) iff gcd(n,k) = 1.

• Theorem 4.3 (Fundamental Theorem of Cyclic Groups): Every subgroup of a cyclic group ⟨a⟩ of order n is cyclic; the order of any subgroup divides n; for each positive divisor d of n, ⟨a^{n/d}⟩ is the unique subgroup of order d.

• Intuition: If G = ⟨a⟩ with |G| = n, then subgroups correspond precisely to divisors of n; there is exactly one subgroup of each order d|n, generated by a^{n/d}.

• Consequences for structure: All cyclic groups of a given order are isomorphic; abstractly, the specific elements do not matter, only the operation.

Subgroups: Tests and Examples

• Subgroup tests provide quick checks to avoid full group axiom verification:

  • Theorem 3.1 (One-Step Subgroup Test): Let G be a group and H ⊆ G be nonempty. If ab^{-1} ∈ H whenever a,b ∈ H, then H is a subgroup of G.

  • Theorem 3.2 (Two-Step Subgroup Test): If H ≠ ∅, closed under the group operation (ab ∈ H for a,b ∈ H), and closed under inverses (a^{-1} ∈ H for a ∈ H), then H ≤ G.

  • Theorem 3.3 (Finite Subgroup Test): If H is a nonempty finite subset of G and closed under the group operation, then H ≤ G.

• Examples of subgroups and non-subgroups illustrate these tests:

  • H = {x ∈ G | x^2 = e} is a subgroup of an Abelian group G (requires closure under products of squares).

  • H = {x ∈ G | |x| is finite} is a subgroup for Abelian G (closure of product of finite-order elements).

  • If G is Abelian and H,K ≤ G, then HK = {hk | h ∈ H, k ∈ K} is a subgroup of G.

  • Non-subgroup examples: H = {x ∈ G | x = 1 or x is irrational} in G = R^× (multiplication) fails closure; K = {x ∈ G | x ≥ 1} fails closure under inverses.

• How to show a subset is not a subgroup: (i) Identity missing; (ii) Inverse missing; (iii) Closure fails for some pair.

• Cyclic subgroups via repetition: kal = {a^n | n ∈ Z} is a subgroup; this is the cyclic subgroup generated by a.

• Examples of cyclic subgroups:

  • In U(10) = {1,3,7,9}, ⟨3⟩ = ⟨7⟩ = {1,3,9,7} (each generates the whole group).

  • In Z_10, ⟨2⟩ = {0,2,4,6,8}.

  • In Z (integers under addition), ⟨1⟩ = Z and ⟨−1⟩ = Z (both generators).

• Subgroups in various groups illustrate that subgroups can have different forms, yet in cyclic groups they are all cyclic by Theorem 4.3.

Cyclic Groups

• Definition: A group G is cyclic if there exists a ∈ G such that G = ⟨a⟩.

• Theorem 4.1 (Criterion for ai = aj):

  • If |a| = ∞, then a^i = a^j iff i = j.

  • If |a| = n, then ⟨a⟩ = {e, a, a^2, …, a^{n−1}} and a^i = a^j iff n | (i − j).

• Corollary 1: For any a ∈ G, |a| ≤ |⟨a⟩|.

• Corollary 2: If a^k = e, then |a| divides k (i.e., |a| | k).

• Theorem 4.2: If |a| = n and k ≥ 1, then ⟨a^k⟩ has order n / gcd(n,k); in particular, |a^k| = n / gcd(n,k).

• Corollary 3: If |a| = n, then ⟨a^j⟩ = ⟨a⟩ iff gcd(n,j) = 1; in particular, a^j is a generator iff gcd(n,j) = 1.

• Corollary 4: In Zn, k is a generator of Zn iff gcd(n,k) = 1.

• Theorem 4.3 (Fundamental Theorem of Cyclic Groups): Every subgroup of a cyclic group is cyclic; if |⟨a⟩| = n, then the order of any subgroup divides n; for each divisor d of n there is exactly one subgroup of order d, namely ⟨a^{n/d}⟩.

• Conceptual takeaway: Cyclic groups serve as prototypes for all cyclic groups; multiplication in ⟨a⟩ mimics addition in Z or Zn depending on whether a has infinite or finite order.

Connections and Historical Context

• Historical note: Non-commutativity of matrix multiplication played a key role in early quantum mechanics (Heisenberg, Born) and the matrix formalism of quantum theory.

• The development of group theory arose from examples and evolved into an axiomatic framework to capture structural properties independent of the specific nature of the elements.

Quick Reference: Key Notation and Concepts

• Group: (G,·) with associativity, identity e, inverses for all elements.

• Subgroup: H ≤ G; H inherits the group operation and satisfies axioms.

• Abelian: ab = ba for all a,b ∈ G.

• ⟨a⟩: cyclic subgroup generated by a; set {a^n | n ∈ Z}.

• Order of a: |a| is the smallest n > 0 with a^n = e (or ∞ if no such n).

• Order of G: |G|, finite or infinite.

• Exponent notations: g^n (n ∈ Z), with g^0 = e and g^{−n} = (g^{-1})^n.

• Corollaries and theorems provide quick rules for orders, generators, and subgroups in cyclic settings.

Exercises (Selective categories to guide study)

• Test closure, associativity, and inverse existence in concrete sets.

• Compute orders of elements in specific finite groups; determine generators of cyclic subgroups.

• Verify subgroup criteria using Theorems 3.1–3.3; construct Cayley tables and verify Latin-square property for group tables.

• Identify subgroups HK in Abelian groups and discuss conditions for non-Abelian cases.

• Apply Theorem 4.2 to compute orders of powers and subgroups in cyclic groups.

• Prove that every subgroup of a cyclic group is cyclic and enumerate subgroups given the order of the group.

• Note: The exercises in the source material include a broad mix of conceptual, computational, and proof-based tasks; refer to the table of contents and examples for guided practice.

Binary Operation

• A binary operation on a set G is a function that assigns to each ordered pair (a,b) \in G \times G an element in G. This implies closure: ab \in G for all a,b \in G.

• Example: Addition on the integers (Z) is a binary operation, as the sum of any two integers is always an integer.

Definition: Group

• A group is a set G together with a binary operation (multiplication) that satisfies three properties:

  1. Associativity: (ab)c = a(bc) for all a,b,c \in G.

  2. Identity: There exists an element e \in G such that ae = ea = a for all a \in G.

  3. Inverses: For each a \in G there exists b \in G (an inverse of a) such that ab = ba = e.

• Example: The set of integers (Z) under ordinary addition is a group. The identity element is 0, and the inverse of any integer a is -a.

Abelian Group

• If ab = ba for all a,b \in G, the group is Abelian (commutative).

• Example: The set of integers (Z) under ordinary addition is an Abelian group because a+b = b+a for all integers a and b.

Cyclic Group

• A group G is cyclic if there exists an element a \in G such that G = \langle a \rangle (meaning all elements in G can be expressed as a power of a).

• Example: The set Z_n = {0,1,2,\dots,n-1} under addition modulo n is a cyclic group, generated by 1 (e.g., 1+1 = 2\pmod n, 1+1+1=3\pmod n, etc.). For instance, U(10) = {1,3,7,9} is cyclic, as \langle 3 \rangle = {3^1=3, 3^2=9, 3^3=27 \equiv 7, 3^4=81 \equiv 1} = U(10).

Subgroup

• A subset H of a group G is a subgroup of G (denoted H \le G) if H is itself a group under the operation inherited from G.

• Example: In Z_{10} = {0,1,2,3,4,5,6,7,8,9} under addition modulo 10, the set \langle 2 \rangle = {0,2,4,6,8} is a subgroup. This is a cyclic subgroup generated by 2.