Alphabeta MathHow 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.
Second isomorphism theorem for groups:
Statement
Second isomorphism theorem for groups: .
If and , then
Facts & Assumptions
Given: A subgroup and a normal subgroup .
and (If and , then is a subgroup and ).
A homomorphism modulo its kernel is isomorphic to its image (First isomorphism theorem for groups: ).
The canonical quotient map has the given normal subgroup as kernel (The canonical projection , , is a surjective group homomorphism).
Quotients by normal subgroups are groups of cosets (The quotient group and coset product ).
Proof
Restrict the quotient map to , ; [L1] and [L4] make this a homomorphism.
Its kernel is , while every shows that its image is .
The kernel and image calculation in step 2.1 gives .
Depends on
- If $H\le G$ and $N\mathrel{\trianglelefteq}G$, then $HN$ is a subgroup and $H\cap N\mathrel{\trianglelefteq}H$
- First isomorphism theorem for groups: $G/\ker f\cong\operatorname{im}f$
- The canonical projection $\pi:G\to G/N$, $\pi(g)=gN$, is a surjective group homomorphism
- The quotient group $G/N$ and coset product $(gN)(hN)=ghN$
Used by
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 35 results over 18 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
- Judson, Abstract Algebra: Theory and Applications, Isomorphism Theorems (standard reference, not scraped)