Finite quotient group

Mathematical statement

First certify N as a normal subgroup of the complete group table. All left cosets form a disjoint complete partition; quotient elements are coset sets, not representatives. Record multiplication by coset indices and verify every replacement of either representative lands in the same product coset. N is the identity coset and g^(-1)N is the inverse. A nonnormal subgroup can have representative-dependent products, blocking this quotient multiplication. Quotients are not automatically commutative, and no first-isomorphism claim is added.

Representative formula (exact TeX)

Q(G,N)=C

Assumptions

• Families use complete finite Cayley tables, an identity and an explicit nonempty subset.
• Coset partitions and every representative pair are exhaustively enumerated.

Symbol scope

• Q — quotient-group operator
• G — finite group
• N — normal subgroup
• C — coset carrier

Intuition, definition, and misconceptions

Build the complete coset partition first.

Quotient multiplication must survive every representative substitution.

A nonnormal witness directly refutes well-definedness.

A representative is not the quotient element itself, and cosets cannot simply be multiplied without normality.

Adopted derivation guide

The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.

Prerequisites

• Finite normal subgroup — Normality is needed for well-defined products.
• Quotient set — Cosets are equivalence classes.
• Finite group — Use the original multiplication and inverses.

Verification boundary

Only the finite quotient construction is made; commutativity or isomorphism with an image is not automatic.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.

Next obligation

Add the first isomorphism theorem for finite groups.

Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.