CounterexampleConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (z-ai/glm-5.2)audited 2026-07-27

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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A nonempty subset of a group closed under the operation need not be a subgroup: the nonnegative integers inside $(\mathbb{Z}, +)$

Statement refuted

False claim: if $G$ is a group and $H \subseteq G$ is nonempty, contains the identity, and is closed under the operation of $G$ , then $H$ is a subgroup of $G$ (Subgroup).

The set of nonnegative integers inside the additive group of $\mathbb{Z}$ refutes it: $H = \{\, x \in \mathbb{Z} : 0 \le x \,\}$ contains $0$ , is closed under addition, and is not a subgroup, because $1 \in H$ while $-1 \notin H$ .

Facts & Assumptions

Given: The abelian group $(\mathbb{Z},+,0)$ (The integers as equivalence classes of pairs of naturals, Arithmetic on the integers, Group and abelian group) and the subset $H = \{\, x \in \mathbb{Z} : 0 \le x \,\}$ , which by The naturals embed in the integers is exactly the image of $\iota : \mathbb{N} \to \mathbb{Z}$ .

[L1]

$\mathbb{Z}$ is a commutative ring; its order is total, antisymmetric and transitive and is compatible with addition (The integers form a commutative ring, The integers form a totally ordered ring, Order on the integers, Arithmetic on the integers).

[L2]

$\iota$ is injective with image exactly the nonnegative integers, and $\iota(0) = 0$ , $\iota(1) = 1$ (The naturals embed in the integers).

[L3]

A subgroup must contain the identity and be closed under the operation and under inverses (Subgroup); equivalently, a nonempty $H$ is a subgroup exactly when $x - y \in H$ for all $x, y \in H$ (One-step subgroup test: a nonempty $H \subseteq G$ is a subgroup iff $gh^{-1} \in H$ for all $g, h \in H$ ; the identity and the inverses of $H$ are then those of $G$ ).

[L4]

A subset closed under an operation inherits it as a binary operation (Binary operation on a set; associativity, commutativity, and a subset closed under the operation).

[L5]

The refuted claim: a nonempty subset of a group containing the identity and closed under the operation is a subgroup.

Counterexample

technique · direct

1.1

$0 \in H$ , since $0 \le 0$ ; so $H$ is nonempty and contains the identity of $(\mathbb{Z},+)$ .

L1given

1.2

$H$ is closed under addition: if $0 \le x$ and $0 \le y$ then $x \le x + y$ by compatibility of the order with addition, and $0 \le x$ , so $0 \le x + y$ by transitivity. Hence $+$ restricts to a binary operation on $H$ , and $(H,+,0)$ is a commutative monoid.

L1L4given

1.3

$0 < 1$ : the integer $1 = \iota(1)$ lies in the image of $\iota$ , so $0 \le 1$ , and $1 \ne 0$ because $\iota$ is injective and $1 \ne 0$ in $\mathbb{N}$ . Hence $1 \in H$ .

L2given

2.1

$-1 \notin H$ : adding $-1$ to both sides of $0 < 1$ gives $-1 < 0$ , so $0 \le -1$ fails by antisymmetry.

step 1.3L1

3.1

Therefore $H$ is not closed under inverses, since $1 \in H$ and its additive inverse $-1$ is not in $H$ ; so $H$ is not a subgroup of $(\mathbb{Z},+)$ .

step 1.3step 2.1L3

4.1

By steps 1.1, 1.2 and 3.1 the set $H$ is a nonempty subset containing the identity and closed under the operation which is not a subgroup, so the claim of [L5] is false.

step 1.1step 1.2step 3.1L5∎

Remarks

Closure under inverses is an independent condition, and this is why the economical criterion of One-step subgroup test: a nonempty $H \subseteq G$ is a subgroup iff $gh^{-1} \in H$ for all $g, h \in H$ ; the identity and the inverses of $H$ are then those of $G$ tests $xy^{-1}$ rather than $xy$ : the single expression carries both closure requirements at once.
For a finite subset the claim would be true, since a nonempty finite subset of a group closed under the operation is a subgroup; the witness above is necessarily infinite. That finiteness result is not proved in the library at this point in the reading order, and nothing here rests on it.
$H$ is a perfectly good commutative monoid, by step 1.2, and $\iota$ is a bijection from $\mathbb{N}$ onto $H$ carrying addition to addition (The naturals embed in the integers). So the example is the same phenomenon as A commutative monoid in which cancellation holds need not be a group: $(\mathbb{N}, +)$ , seen from inside a group.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 58 results over 20 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

Subgroup (Wikipedia) (standard reference, not scraped)
Submonoid (Wikipedia) (standard reference, not scraped)
Cancellation property (Wikipedia) (standard reference, not scraped)

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