What is an Abelian group, from a linear/abstract algebra perspective?

1 Answer
Apr 27, 2016

Answer:

An Abelian group is a group with the additional property of the group operation being commutative.

Explanation:

A group # < G , •> # is a set #G# together with a binary operation #•:GxxG->G# which fulfill the following conditions:

  1. #G# is closed under #•#.
    For any #a,binG#, we have #a•b in G#

  2. #•# is associative.
    For any #a,b,cinG#, we have #(a•b) • (c) = a •(b•c)#

  3. #G# contains an identity element
    There exists #einG# such that for all #ainG#, #a•e=e•a=a#

  4. Each element of #G# has an inverse in #G#
    For all #ainG# there exists #a^(-1)inG# such that #a•a^(-1)=a^(-1)•a=e#

A group is said to be Abelian if it also has the property that #•# is commutative, that is, for all #a,binG#, we have #a•b = b•a#.

The group #< ZZ, +># (the integers with standard addition) is an Abelian group, as it fulfills all five of the above conditions.

The group #GL_2(RR)# (the set of invertible #2"x"2# matrices with real elements together with matrix multiplication) is non-Abelian, as while it fulfills the first four conditions, matrix multiplication between invertible matrices is not necessarily commutative. For example:

#((1,1),(1,0))((1,0),(1,1)) = ((2,1),(1,0))#

but

#((1,0),(1,1))((1,1),(1,0)) = ((1,1),(2,1))#