homogeneous group

A homogeneous group is a set $G$ together with a map $():G\times G\times G\to G$ satisfying:

i)$(a,a,b)=b$

ii)$(a,b,b)=a$

iii)$((a,b,c),d,e)=(a,b,(c,d,e))$

for all $a,b,c,d,e\in G$.

A map $f:G\to H$ of homogeneous groups is a homomorphism if it $f(a,b,c)=(fa,fb,fc)$, for all $a,b,c\in G$.

A non-empty homogeneous group is essentially a group, as given any $x\in G$, we may define the following product on $G$:

$ab=(a,x,b)$.

This gives $G$ the of a group with identity $x$. The choice of $x$ does not affect the isomorphism class of the group obtained.

One may recover a homogeneous group from a group obtained this way, by setting

$(a,b,c)=a{b}^{-1}c$.

Also, every group may be obtained from a homogeneous group.

Homogeneous groups are homogeneous: Given $a,b\in G$ we have a homomorphism $f$ taking $a$ to $b$, given by $fx=(x,a,b)$.

In this way homogeneous groups differ from groups, as whilst often used to describe symmetry, groups themselves have a distinct element: the identity.

Also the definition of homogeneous groups is given purely in of identities, and does not exclude the empty set.

Title	homogeneous group
Canonical name	HomogeneousGroup
Date of creation	2013-03-22 16:12:12
Last modified on	2013-03-22 16:12:12
Owner	whm22 (2009)
Last modified by	whm22 (2009)
Numerical id	15
Author	whm22 (2009)
Entry type	Definition
Classification	msc 20A05
Defines	homomorphism of homogeneous groups