Description: Define class of all
groups. A group is a monoid (df-mnd 16537) whose
internal operation is such that every element admits a left inverse
(which can be proven to be a two-sided inverse). Thus, a group  is
an algebraic structure formed from a base set of elements (notated
    per df-base 15126) and an internal group operation
(notated  per df-plusg 15203). The operation combines any
two elements of the group base set and must satisfy the 4 group axioms:
closure (the result of the group operation must always be a member of
the base set, see grpcl 16679), associativity (so
   for any a, b, c, see
grpass 16680), identity (there must be an element   such
that for
any a), and inverse (for each element a
in the base set, there must be an element   in the base set
such that ).
It can be proven that the identity
element is unique (grpideu 16682). Groups need not be commutative; a
commutative group is an Abelian group (see df-abl 17433). Subgroups can
often be formed from groups, see df-subg 16814. An example of an (Abelian)
group is the set of complex numbers  over the group operation
 (addition),
as proven in cnaddablx 17506; an Abelian group is a group
as proven in ablgrp 17435. Other structures include groups, including
unital rings (df-ring 17782) and fields (df-field 17978). (Contributed by
NM, 17-Oct-2012.) (Revised by Mario Carneiro,
6-Jan-2015.) |
