Lemma·L30
All cosets have the size of the subgroup
Translation is a bijection: every coset is a perfect copy of H.
For every subgroup
and every
:
with
the equinumerosity and
the left coset.
In words
For every subgroup H of a group G and every element a of G, the left coset aH is equinumerous with the subgroup H itself: multiplying by a merely relabels the elements.
Never needed: F08 · F10 · F11 · F12 · A03 · A04 · A05 · A07 · A08 (computed from the citation graph, not asserted).
Proof
- 1Define with domain by : as a set, by Separation, and it is a function since each yields exactly one value (L01), landing in by D040.
- 2
- 3
- 4
∎
Remarks
The whole content of "translation by
is invertible", concentrated into one bijection; the inverse translation is by
. This uniformity is remarkable: however lopsided a subgroup might look inside its group, every coset is an exact-size copy of it. It is one of the two pillars of Lagrange's theorem, the other being the partition into cosets.