Alphabeta Math

Monoids, Groups and Subgroups: Examples and Counterexamples

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✓ 11 results · all verified · 9 also independently AI-judged
Every result on this page is machine-checked by a proof checker and read in full and owner-audited; the judge is an additional, independent cross-model AI review of the proofs. The 2 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.
1 · Prerequisites
Abstract Algebra
Real Analysis
Set Theory
2 · Logical flowchart

3 · Items

Example

For any field F, (F, +) and (F ∖ {0}, ·) are abelian groups; in particular (ℚ, +), (ℚ ∖ {0}, ·), (ℝ, +) and (ℝ ∖ {0}, ·)

Example

(ℤ, +) is an abelian group, (ℤ, ·) is a commutative monoid that is not a group, and its group of units is {1, -1}

Counterexample

A commutative monoid in which cancellation holds need not be a group: (ℕ, +)

Example

For n ≥ 1 the congruence classes modulo n form an abelian group (ℤ/n, +) of order n, generated by the class of 1

Example

Sym({1,2,3}) has exactly six elements, is non-abelian, and its elements have orders 1, 2 and 3

Counterexample

(gh)ⁿ = gⁿhⁿ fails without commutativity: two transpositions in Sym({1,2,3}) with (gh)² ≠ g²h²

Example

The Klein four-group as the subgroup {id, (12)(34), (13)(24), (14)(23)} of Sym({1,2,3,4}): abelian of order 4, non-cyclic, every non-identity element of order 2

Example

The eight vertex permutations of a square form a non-abelian subgroup of Sym({1,2,3,4}) of order 8, generated by a 4-cycle and one diagonal swap

Example

nℤ is a subgroup of (ℤ, +) for every n ∈ ℤ, and every subgroup of (ℤ, +) has this form

Counterexample

A nonempty subset of a group closed under the operation need not be a subgroup: the nonnegative integers inside (ℤ, +)

False statement

FALSE: The union of two subgroups is a subgroup

Sources