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The four cosets of $4 Z$ in $(Z, +)$ reproduce addition modulo $4$

Example

The quotient group $Z / 4 Z$ has the four cosets

$4 Z, 1 + 4 Z, 2 + 4 Z, 3 + 4 Z,$

and its operation is

$(a + 4 Z) + (b + 4 Z) = (a + b) + 4 Z .$

Under the identification of $a + 4 Z$ with the residue class $[a]_{4}$ , this is addition modulo $4$ .

Facts & Assumptions

Given: The additive group $(Z, +)$ and its subgroup $4 Z$ .

[L1]

Every congruence class modulo $4$ has a unique representative among $0, 1, 2, 3$ (For $n \geq 1$ , every class in $Z / n$ has one representative $r$ with $0 \leq r < n$ , so $∣ Z / n ∣ = n$ ; while $Z / 0$ is in bijection with $Z$ ).

[F1]

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

[L2]

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

Verification

technique · direct

1.1

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

1.2

Quotient addition adds representatives, so the sum of $a + 4 Z$ and $b + 4 Z$ is $(a + b) + 4 Z$ .

2.1

Sending $a + 4 Z$ to $[a]_{4}$ 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

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

20 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

UCL lecture notes, Normal subgroups and quotients (standard reference, not scraped)

The four cosets of 4Z4Z in (Z,+)(Z,+) reproduce addition modulo 44