Kernel, fibers and injectivity from C4 to C2

Goal and scope

For f:C4 to C2, f(a)=a mod 2, the normal kernel is {0,2}; fibers are its cosets and f is not injective.

Two complete additive-group tables and a total homomorphism, without quotient or first-isomorphism construction.

Givens and symbols

• calc

• G1 — G={0,1,2,3} under addition modulo 4, Q={0,1} under addition modulo 2; f's aligned table is [0,1,0,1].

• K — source kernel subgroup
• f — complete total homomorphism

Rules and sources

• calc · prerequisite — Assume exact modular integer arithmetic and definitions of finite groups, homomorphisms and preimages; use additive notation here.
• mko_definition · cited — Both groups require complete Cayley-table qualification. f is a total mapping and must satisfy f(x*y)=f(x)*f(y) for every pair. Its kernel is the entire target-identity preimage, with subgroup and all-g conjugation closure checked directly. f(a)=f(b) exactly when a^(-1)*b belongs to the kernel, so each nonempty fiber is a kernel coset. The kernel is the source-identity singleton exactly when f is injective. Nonhomomorphisms cannot invoke these theorems; quotient/image isomorphism is not claimed here. · mko-finite-group-homomorphism-kernel

Main derivation

G=C_4,\quad Q=C_2

\begin{array}{c|cccc}+&0&1&2&3\\\hline 0&0&1&2&3\\1&1&2&3&0\\2&2&3&0&1\\3&3&0&1&2\\\end{array}\qquad\begin{array}{c|cc}+&0&1\\\hline 0&0&1\\1&1&0\\\end{array}

f:G\to Q

f(0)=0,f(1)=1,f(2)=0,f(3)=1

f((a+b)\bmod4)=((a+b)\bmod4)\bmod2

\begin{array}{c|cccc}f(a+b)&0&1&2&3\\\hline 0&0&1&0&1\\1&1&0&1&0\\2&0&1&0&1\\3&1&0&1&0\\\end{array}\quad=f(a)+f(b)\ (\bmod2)

K=f^{-1}(\{0_Q\})

K=\{0,2\}

K=\{0,2\}

0\in K;\quad0+0=0,0+2=2,2+0=2,2+2=0;\quad-0=0,-2=2\ (\bmod4)

g=0,1,2,3,\quad k=0,2

\begin{array}{c|cc}g+k-g&0&2\\\hline0&0&2\\1&0&2\\2&0&2\\3&0&2\end{array}

f^{-1}(0)=\{0,2\},\quad f^{-1}(1)=\{1,3\}

f(a)=f(b)\iff b-a\in K;\quad f^{-1}(0)=0+K,\ f^{-1}(1)=1+K

0\ne2,\quad f(0)=f(2)=0

f\text{ not injective},\quad K\ne\{0\}

f(a)=f(b)\iff b-a\in K

K=\{0\}\iff f\text{ injective}

Substeps and lemmas

No substeps in this guide.

Reverse audits

• S09 · premise_audit · G1, S03, S04, S07, S08 — The equivalence agrees with both conditions failing here and adds no quotient isomorphism.
• S08 · premise_audit · G1, S02, S04, S07 — One collision suffices; being onto Q would not imply injectivity.
• S07 · inverse_equivalence · G1, S01, S04, S06 — The additive difference b-a corresponds to a inverse b in multiplicative notation, using the same source operation.
• S06 · premise_audit · G1, S01, S03, S05 — The model is a qualified homomorphism; an arbitrary map's identity preimage need not be a subgroup.
• S05 · premise_audit · G1, S01, S04 — The inverse in additive notation is negation modulo 4, not a rational reciprocal.
• S04 · premise_audit · G1, S02, S03 — Kernel elements belong to source G, not the target value set.
• S03 · premise_audit · G1, S01, S02 — This checks all pairs, not just generators, using both the formula and 16-entry table.
• S02 · premise_audit · G1, S01 — A total map is not yet a homomorphism; preservation remains to be checked.
• S01 · premise_audit · G1 — Qualify both groups before applying kernel facts to a mapping array.

Error controls and negative controls

Target identity

The kernel is the entire preimage of target 0, not merely the source label 0.

Homomorphism first

Check all 16 operation pairs before kernel conclusions.

• Any total map has a normal identity preimage. → reject — The map C4 to C2 with array [0,0,1,1] has identity preimage {0,1}, not closed since 1+1=2, and is not a homomorphism.
• K={0}. → reject — f(2)=0 puts 2 in the kernel too.
• f is surjective, hence injective. → reject — f(0)=f(2) is a source collision.

Result and limitations

For f:C4 to C2, f(a)=a mod 2, the normal kernel is {0,2}; fibers are its cosets and f is not injective.

Verification and source binding

1 declared checks for this guide; only the listed expressions are covered.

• source commit: b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f
• adopted source SHA-256: bcd46de77dc90f619f8492cc26c9e5e506e7f2107662144b4c6d8ff101c244b0
• original source SHA-256: bcd46de77dc90f619f8492cc26c9e5e506e7f2107662144b4c6d8ff101c244b0
• Lean: NOT_RUN · independent AI: SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW

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