Groups, Subgroups, Cyclic Groups and Orders
Handbook guide to groups, subgroups, cyclic groups and orders with core definitions, structural results, reasoning methods and verification checks.
How the topic fits together
Groups and Subgroups
This source section supplies the definitions, constructions and formal results used in this article. The sequence of results is preserved so later claims are not detached from their prerequisites.
Core definitions and structural results
The following cards retain the mathematical content of the source while condensing long proofs into verification strategies. Numerical examples are treated as examples, not universal requirements.
Definition
A group is a nonempty set G on which there is defined a binary operation (a, b) →ab satisfying the following properties: Closure: If a and b belong to G, then ab is also in G; Associativity: a(bc) = (ab)c for all a, b, c ∈G; Identity: There is an element 1 in G such that a1 = 1a = a for all a in G; Inverse: If a is in G there is an element a−1 in G such that aa−1 = a−1a = 1. A group G is abelian if the binary operation is commutative, i.e., ab = ba for all a, b in G. In this case the binary operation is often written additively ((a, b) →a + b)), with the identity written as 0 rather than 1. There are some very familiar examples of abelian groups under addition, namely the integers Z, the rationals Q, the real numbers R, the complex numbers C, and the integers Zm modulo m.
Proof / verification strategy: Use this as a definition checklist: identify the ambient object, test each stated condition separately, then compare examples and non-examples without assuming later theorems.
Definitions and Comments
A subgroup H of a group G is a nonempty subset of G that forms a group under the binary operation of G. Equivalently, H is a nonempty subset of G such that if a and b belong to H, so does ab−1. (Note that 1 = aa−1 ∈H; also ab = a((b−1)−1) ∈H.) If A is any subset of a group G, the subgroup generated by A is the smallest subgroup containing A, often denoted by < A >. Formally, < A > is the intersection of all subgroups containing A.
Proof / verification strategy: Use this as a definition checklist: identify the ambient object, test each stated condition separately, then compare examples and non-examples without assuming later theorems.
Definitions and Comments
The groups G1 and G2 are said to be isomorphic if there is a bijection f : G1 →G2 that preserves the group operation, in other words, f(ab) = f(a)f(b). Isomorphic groups are essentially the same; they differ only notationally. Here is a simple example. A group G is cyclic if G is generated by a single element: G =< a >.
Proof / verification strategy: Use this as a definition checklist: identify the ambient object, test each stated condition separately, then compare examples and non-examples without assuming later theorems.
Proposition
If G is a finite cyclic group of order n, then G has exactly one (necessarily cyclic) subgroup of order n/d for each positive divisor d of n, and G has no other subgroups. If G is an infinite cyclic group, the (necessarily cyclic) subgroups of G are of the form {1, b, b2, ...}, where b is an arbitrary element of G, or in additive notation, {0, b, 2b, ...}.
Proof / verification strategy: Use subgroup generation, coset/orbit counting and divisibility of orders. For cyclic groups, reduce the claim to arithmetic on exponents modulo the group order.
Proposition
If G is a cyclic group of order n generated by a, the following conditions are equivalent: (a) |ar| = n. (b) r and n are relatively prime. (c) r is a unit mod n, in other words, r has an inverse mod n (an integer s such that rs ≡1 mod n). Furthermore, the set Un of units mod n forms a group under multiplication.
Proof / verification strategy: Use subgroup generation, coset/orbit counting and divisibility of orders. For cyclic groups, reduce the claim to arithmetic on exponents modulo the group order.
Quick-reference relationships
Problem-solving workflow
Identify the ambient group
State the operation, identity, inverses and whether commutativity is available.
Locate the relevant subgroup structure
Check generated subgroups, normality, cosets, stabilisers or direct factors before applying a theorem.
Use the correct counting or mapping tool
Choose coset counting, orbit counting, a homomorphism, a quotient or a group action as appropriate.
Check hypotheses explicitly
Finite-order assumptions, normality, prime-power divisibility and coprimality conditions are not interchangeable.
Translate the result back to structure
Interpret the result as a statement about subgroup size, quotient structure, action, decomposition or presentation.
Verify with a small model
Use a cyclic, symmetric, dihedral or modular example to test the logic without treating the example as the theorem.
Worked-solution emphasis from the supplied source
The supplied worked solutions for this section repeatedly test order, subgroup, matrix, prime, factor, Tor. These checks are used here as verification themes rather than copied as answer text.
Common mistakes and boundary conditions
- Treating left and right cosets as identical without normality.
- Assuming the converse of a subgroup-order divisibility result.
- Confusing the order of a group with the order of one of its elements.
- Using quotient multiplication before checking that the subgroup is normal.
Verification checklist
- State the ambient algebraic structure and operation before applying a theorem.
- Record every hypothesis that controls the result: finiteness, commutativity, normality, primality, separability, exactness or other section-specific conditions.
- Distinguish a definition from a theorem that follows from it.
- Check whether a map is well-defined before using its kernel, image, inverse or induced map.
- Use a concrete example only as a check; do not promote an illustrative value or pattern to a universal rule.
- When a quotient, localisation or extension is constructed, identify the canonical map and what becomes equal, invertible or fixed.
Source coverage map
| Source section | Subject | PDF pages analysed |
|---|---|---|
| 1.1 | Groups and Subgroups | 10–11 |
Related Mathematics pages
Source note: synthesised from the supplied abstract-algebra PDF. The complete 298-page file, including diagrams and worked solutions, was reviewed. Source-identifying author and bibliographic personal details are intentionally omitted. Formal proofs are condensed; the page does not claim requirements or values not supported by the supplied mathematics.