Definition·D040
Coset
A subgroup shifted by a group element.
the left coset of the subgroup
by the element
.
In words
The left coset of H by a is the set of all products of a with members of H: H translated by a.
Rests onA02
Never needed: F02 · F03 · F04 · F05 · F06 · F08 · F09 · F10 · F11 · F12 · F13 · F14 · A01 · A03 · A04 · A05 · A06 · A07 · A08 (computed from the citation graph, not asserted).
Remarks
Exists by Separation. Basic instances:
, and
always, since
with
(D039, L28). Distinct-looking cosets can coincide:
exactly when
, which is the coset equivalence relation. All cosets have the same size as
itself (L30), and together they tile the group, which is the geometric content of Lagrange's theorem. Right cosets
are defined symmetrically and tell the same story.
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