**Description: **Define class of all
groups. A group is a monoid (df-mnd 14330) 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 13116) and an internal group operation
(notated per df-plusg 13184). 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 14458), associativity (so
for any a, b, c, see
grpass 14459), 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 14461). Groups need not be commutative; a
commutative group is an Abelian group (see df-abl 15055). Subgroups can
often be formed from groups, see df-subg 14581. An example of an (Abelian)
group is the set of complex numbers over the group operation
(addition),
as proven in cnaddablx 15121; an Abelian group is a group
as proven in ablgrp 15057. Other structures include groups, including
unital rings (df-ring 15303) and fields (df-field 15478). (Contributed by
NM, 17-Oct-2012.) (Revised by Mario Carneiro,
6-Jan-2015.) |