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.
If is finite then ; for finite this equals
Statement
Let . If is finite, then the quotient group is finite and
In particular, if is finite, then
Facts & Assumptions
Given: A group and a normal subgroup .
The index is the finite cardinality of the left-coset set when that set is finite (The coset set and the index of a subgroup).
If is finite and , then (Lagrange's theorem: for every subgroup of a finite group ).
The order of a finite group is the cardinality of its underlying set (The order of a finite group and the order of an element, with when no positive power of is the identity).
The quotient group has the left cosets of as its underlying set (For , the cosets form a group with identity and inverse ).
Proof
If is finite, then by [F1] the coset set underlying is finite with cardinality ; hence [F2] and [L2] give .
If is finite, then [L1] gives . Since contains the identity, , and step 1.1 yields .
The two asserted formulas follow.
Depends on
- For $N\mathrel{\trianglelefteq}G$, the cosets form a group with identity $N$ and inverse $(gN)^{-1}=g^{-1}N$
- The coset set $G/H$ and the index $[G:H]$ of a subgroup
- Lagrange's theorem: $|G|=[G:H]|H|$ for every subgroup $H$ of a finite group $G$
- The order $|G|$ of a finite group and the order $\operatorname{ord}(g)$ of an element, with $\operatorname{ord}(g) = \infty$ when no positive power of $g$ is the identity
Used by
- Converse of Lagrange for finite abelian groups: every divisor occurs as a subgroup order Corollary
- Every group of order p², for prime p, is abelian Corollary
- The three-cycle subgroup of Sym({1,2,3}) is normal and its quotient has two elements Example
- A nontrivial finite abelian p-group with a unique subgroup of order p is cyclic Lemma
- The successive quotients pⁱG/pⁱ⁺¹G recover the cyclic summand multiplicities of a finite abelian p-group Lemma
- A maximal-order cyclic subgroup splits off a finite abelian p-group Theorem
- A p-primary component has the full p-power order and is the unique subgroup of that order Theorem
- Cauchy's theorem for finite abelian groups Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 75 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
- T. W. Judson, Abstract Algebra: Theory and Applications, Factor Groups and Normal Subgroups (standard reference, not scraped)