Definition·D038
Group
A set with an associative operation, an identity, and inverses.
A group is an ordered pair
where
is a binary operation on
satisfying
Such an
is an identity (the identity clause is (G2)); a
with
is an inverse of
(the inverse clause is (G3)).
In words
A group is a set with an operation that is associative, and has an identity element and gives every element an inverse.
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
The identity and the inverses are provably unique (L28), so one speaks of the identity
and writes
for the inverse of
. If moreover
for all
, the group is abelian. Examples: the one-element group
; the bijections of any set
under composition, with identity
and inverses from L03 (the symmetric group of
, associativity noted in D018). Non-example:
has an identity but no inverses (D027); the integers would repair this, but are not constructed in this library yet. Groups are the algebra of symmetry; the first structural theorem about finite ones is Lagrange's theorem.
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