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The four cosets of 4Z4\mathbb Z in (Z,+)(\mathbb Z,+) reproduce addition modulo 44

Example

The quotient group Z/4Z\mathbb Z/4\mathbb Z has the four cosets

4Z,1+4Z,2+4Z,3+4Z,4\mathbb Z,\quad 1+4\mathbb Z,\quad 2+4\mathbb Z,\quad 3+4\mathbb Z,

and its operation is

(a+4Z)+(b+4Z)=(a+b)+4Z.(a+4\mathbb Z)+(b+4\mathbb Z)=(a+b)+4\mathbb Z.

Under the identification of a+4Za+4\mathbb Z with the residue class [a]4[a]_4, this is addition modulo 44.

Facts & Assumptions

Given: The additive group (Z,+)(\mathbb Z,+) and its subgroup 4Z4\mathbb Z.

[L1]

Every congruence class modulo 44 has a unique representative among 0,1,2,30,1,2,3 (For n≥1n\ge 1, every class in Z/n\mathbb{Z}/n has one representative rr with 0≤r<n0\le r<n, so ∣Z/n∣=n\lvert\mathbb{Z}/n\rvert=n; while Z/0\mathbb{Z}/0 is in bijection with Z\mathbb{Z}).

[F1]

The integers modulo 44 are congruence classes, and addition is defined by [a]4+[b]4=[a+b]4[a]_4+[b]_4=[a+b]_4 (The congruence class [a]n[a]_n and the quotient set Z/n\mathbb{Z}/n, Addition and multiplication on Z/n\mathbb{Z}/n by [a]n+[b]n=[a+b]n[a]_n+[b]_n=[a+b]_n and [a]n[b]n=[ab]n[a]_n[b]_n=[ab]_n).

[L2]

The quotient group Z/4Z\mathbb Z/4\mathbb Z is literally the same set of classes with the same addition as the additive group of integers modulo 44 (For every n∈Nn\in\mathbb N, the congruence-class group (Z/n,+)(\mathbb Z/n,+) is the quotient group (Z,+)/nZ(\mathbb Z,+)/n\mathbb Z).

Verification

technique · direct
1.1

By [L1], every coset a+4Za+4\mathbb Z equals exactly one of the four displayed cosets, and the four are distinct.

L1
1.2

Quotient addition adds representatives, so the sum of a+4Za+4\mathbb Z and b+4Zb+4\mathbb Z is (a+b)+4Z(a+b)+4\mathbb Z.

L2
2.1

Sending a+4Za+4\mathbb Z to [a]4[a]_4 therefore matches the four cosets with the four residue classes and carries the operation in step 1.2 to the modular addition in [F1].

step 1.1step 1.2F1L2∎

Depends on

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