Alphabeta MathGroup Homomorphisms and the Isomorphism Theorems: Examples and Counterexamples
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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
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✓ 5 results · all verified · 0 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 5 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.
1 · Prerequisites
Abstract Algebra
Number Theory
Real Analysis
2 · Logical flowchart
3 · Items
Example
For n≥2, reduction ℤ→ℤ/n has kernel nℤ and realises ℤ/n by the first isomorphism theorem
Counterexample
The doubling endomorphism of (ℤ,+) has trivial kernel but is not surjective
Example
The trivial homomorphism G→ H has kernel G and image {e_H}
Example
Every inner automorphism of an abelian group is the identity
Example