Description: Define class of all
groups. A group is a monoid (df-mnd 16530) 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 15119) and an internal group operation
(notated  per df-plusg 15196). 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 16672), associativity (so
   for any a, b, c, see
grpass 16673), 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 16675). Groups need not be commutative; a
commutative group is an Abelian group (see df-abl 17426). Subgroups can
often be formed from groups, see df-subg 16807. An example of an (Abelian)
group is the set of complex numbers  over the group operation
 (addition),
as proven in cnaddablx 17499; an Abelian group is a group
as proven in ablgrp 17428. Other structures include groups, including
unital rings (df-ring 17775) and fields (df-field 17971). (Contributed by
NM, 17-Oct-2012.) (Revised by Mario Carneiro,
6-Jan-2015.) |
