mixed group

A mixed group is a partial groupoid G such that G contains a non-empty subset K, called the kernel of G, with the following conditions:

  1. 1.

    if a,bG, then ab is defined iff aK,

  2. 2.

    if a,bK and cG, then (ab)c=a(bc),

  3. 3.

    if aK, then KaKKa,

  4. 4.

    if aK and bG such that ab=b, then ac=c for all cG.

Mixed groups are generalizationsPlanetmathPlanetmath of groups, as the following propositionPlanetmathPlanetmathPlanetmath illustrates:

Proposition 1.

If K=G, then G is a group.


G is a groupoidPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath by condition 1, and a semigroup by condition 2.

Now, by condition 3, given aG, there is bG such that ba=a, so that bc=c for all cG by condition 4. In other words, b is a left identityPlanetmathPlanetmath of G. Again, by condition 3, for every aG, there is a dG such that b=da. So ad=a(bd)=a(da)d=(ad)2, so, by condition 4, adx=x for all xG. In particular, set x=a, we get a=(ad)a=a(da)=ab. Hence, b is a two-sided identityPlanetmathPlanetmath, and G is a monoid.

Finally, by condition 3, for every aG, there are c,dG, such that b=ac=da. So, c=bc=(da)c=d(ac)=db=d, showing that a has a two-sided inverseMathworldPlanetmathPlanetmathPlanetmathPlanetmath. This means that G is a group. ∎

For a non-trivial example of a mixed group, let G be a group and H a subgroupMathworldPlanetmathPlanetmath of G. Define a new multiplicationPlanetmathPlanetmath on G as follows: ab is defined iff aH, and if ab is defined, it is defined as ab, the group multiplication of a and b. Then (G,) is a mixed group. Clearly, associativity of is automatically satisfied. Next, pick any aH, then, for any bH, a-1b and ba-1 are both elements of H, so that baHHa, and condition 3 is also satisfied. Finally, if aH and bG such that ab=b, then a is the multiplicative identityPlanetmathPlanetmath of G, clearly ac=c for all cG.


  • 1 R. H. Bruck, A Survey of Binary Systems, Springer-Verlag, 1966
  • 2 R. Baer, Zur Einordnung der Theorie der Mischgruppen in die Gruppentheorie, S.-B. Heidelberg. Akad. Wiss., Math.-naturwiss. KI. 1928, 4, 13 pp
  • 3 R. Baer, Über die Zerlegungen einer Mischgruppe nach einer Untermischgruppe, S.-B. Heidelberg. Akad. Wiss., Math.-naturwiss. KI. 1928, 5, 13 pp
  • 4 A. Loewy, Über abstrakt definierte Transmutationssysteme oder Mischgruppen, J. reine angew. Math. 157, pp 239-254, 1927
Title mixed group
Canonical name MixedGroup
Date of creation 2013-03-22 18:42:32
Last modified on 2013-03-22 18:42:32
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 7
Author CWoo (3771)
Entry type Definition
Classification msc 20N99
Defines kernel