View source for Amalgam of groups

A family of groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012120/a0121201.png" />, that satisfies the condition that the intersection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012120/a0121202.png" /> is a subgroup in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012120/a0121203.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012120/a0121204.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012120/a0121205.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012120/a0121206.png" />. An example of an amalgam of groups is an arbitrary family of subgroups of a given group. An imbedding of an amalgam of groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012120/a0121207.png" /> into a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012120/a0121208.png" /> is a one-to-one mapping of the union <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012120/a0121209.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012120/a01212010.png" /> whose restriction to each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012120/a01212011.png" /> is an isomorphism. An amalgam of groups in which all intersections <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012120/a01212012.png" /> are identical (and equal to, say, a subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012120/a01212013.png" />) is imbeddable in the group that is the free product of the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012120/a01212014.png" /> with the amalgamated subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012120/a01212015.png" />. On the other hand, there exists an amalgam of four Abelian groups that is not imbeddable in a group. The principal problem concerning amalgams of groups is, generally speaking, as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012120/a01212016.png" /> be possible properties of groups. The question to be answered is the nature of the conditions under which an amalgam of groups with the property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012120/a01212017.png" /> is imbeddable in a group with the property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012120/a01212018.png" />. It was found that all amalgams of two finite groups are imbeddable in a finite group. An amalgam of three Abelian groups is imbeddable in an Abelian group. An amalgam of four Abelian groups imbedded in a group is contained in an Abelian group. There exists an amalgam of five Abelian groups which is imbeddable in a group, but not in an Abelian group. Another problem that has been studied is the imbeddability of an amalgam of groups if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012120/a01212019.png" /> denote solvability, nilpotency, periodicity, local finiteness, etc. (in different combinations).



====Comments====
In the definition of amalgam above, think of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012120/a01212020.png" /> as all being subsets of some large set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012120/a01212021.png" />. The amalgamated product of groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012120/a01212022.png" />  "over a common subgroup U"  is constructed as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012120/a01212023.png" /> be a set of groups indexed by the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012120/a01212024.png" />. For each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012120/a01212025.png" /> let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012120/a01212026.png" /> be a subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012120/a01212027.png" /> and for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012120/a01212028.png" /> let there be an isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012120/a01212029.png" /> identifying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012120/a01212030.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012120/a01212031.png" />. Consider the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012120/a01212032.png" /> of all words

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012120/a01212033.png" /></td> </tr></table>

with each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012120/a01212034.png" /> from some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012120/a01212035.png" />, and consider the following elementary equivalences

1) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012120/a01212036.png" /> then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012120/a01212037.png" /> is equivalent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012120/a01212038.png" />;

2) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012120/a01212039.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012120/a01212040.png" /> belong to the same group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012120/a01212041.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012120/a01212042.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012120/a01212043.png" /> then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012120/a01212044.png" /> is equivalent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012120/a01212045.png" />;

3) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012120/a01212046.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012120/a01212047.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012120/a01212048.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012120/a01212049.png" /> is equivalent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012120/a01212050.png" />.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012120/a01212051.png" /> be the equivalence relation generated by these elementary equivalences, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012120/a01212052.png" /> is the amalgamated product of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012120/a01212053.png" />, more precisely of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012120/a01212054.png" /> (i.e. the free product with amalgamated subgroup of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012120/a01212055.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012120/a01212056.png" />); the group law is induced by concatenation.

Amalgamated products are non-trivial. This follows from the following canonical form theorem. For each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012120/a01212057.png" /> select a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012120/a01212058.png" /> of left coset representatives of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012120/a01212059.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012120/a01212060.png" />. Then each word is equivalent to precisely one of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012120/a01212061.png" /> with each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012120/a01212062.png" /> in some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012120/a01212063.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012120/a01212064.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012120/a01212065.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012120/a01212066.png" /> belonging to different <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012120/a01212067.png" />'s for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012120/a01212068.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012120/a01212069.png" /> one obtains of course the free product of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012120/a01212070.png" />. A subgroup of a free product is itself a free product (Kurosh' theorem). Subgroups of a product with an amalgamated subgroup need not be themselves of this type. The reason is that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012120/a01212071.png" /> is the amalgamated subgroup, then one can take subgroups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012120/a01212072.png" /> of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012120/a01212073.png" /> with different intersections with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012120/a01212074.png" /> so that the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012120/a01212075.png" /> will amalgamate in various different ways. This leads to generalized amalgamated products and the notion of amalgam as defined above. The theory of these is still incomplete.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Hall jr.,   "The theory of groups" , Macmillan  (1959)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H. Neumann,   "Generalized free products with amalgamated subgroups I"  ''Amer. J. Math.'' , '''70'''  (1948)  pp. 590–625</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  H. Neumann,   "Generalized free products with amalgamated subgroups II"  ''Amer. J. Math.'' , '''71'''  (1949)  pp. 491–540</TD></TR></table>