CorollaryStatement: AI-adaptedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-11

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The center of a free product with at least two nontrivial factors is trivial

Statement

If at least two factors in a free product are nontrivial, then its center is the trivial subgroup.

Facts & Assumptions

Given: The objects and hypotheses in the statement.

[L1]

Let $G$ be a group (def-group). The center of $G$ is $Z(G):=\{z\in G:zg=gz\text{ for every }g\in G\}.$ Thus $Z(G)$ consists of the elements that commute with every element of $G$ . Its subgroup and normality properties are proved in lem-center-is-normal. (The center $Z(G)$ of a group).

[L2]

For groups as in def-group, a syllable is a tagged pair $(i,g)$ with $i\in I$ and $g\in G_i\setminus\{e_i\}$ . A reduced syllable word is a finite list of syllables, indexed by a natural length as in def-natural-numbers, in which adjacent tags differ. The empty list is allowed. At a concatenation seam, adjacent syllables from the same factor are multiplied and an identity result is deleted; this elementary reduction is repeated until the seam is reduced. (Reduced syllable words in a family of groups).

[L3]

Every element of $\ast_{i\in I}G_i$ has a unique reduced syllable expression. The identity is represented by the empty word, and no nonempty reduced word represents the identity. (Normal form theorem for free products).

Proof

technique · contradiction

1.1

Assume for contradiction that a nonidentity central element has reduced word $z=x_1\cdots x_n$ . Choose a nonidentity syllable $g$ from a factor different from the factor of $x_1$ .

assume-contragivenL1L2L3

2.1

Then $gz$ is reduced of length $n+1$ . If the last syllable of $z$ lies in the factor of $g$ , the word $zg$ reduces to length at most $n$ ; otherwise it is reduced of length $n+1$ but begins in a different factor from $gz$ .

step 1.1

3.1

Normal-form uniqueness gives $gz\ne zg$ in either case, contradicting centrality. Hence only the identity is central.

step 2.1discharge-contradiction∎

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: 19 results over 10 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

Allen Hatcher, Algebraic Topology, Ch. 1, §1.2, Exercise 1 (standard reference, not scraped)