# Amalgam of groups

A family of groups $G _ \alpha , \alpha \in I$, that satisfies the condition that the intersection $G _ \alpha \cap G _ \beta$ is a subgroup in $G _ \alpha$ and $G _ \beta$ for any $\alpha , \beta$ from $I$. An example of an amalgam of groups is an arbitrary family of subgroups of a given group. An imbedding of an amalgam of groups $A = \{ {G _ \alpha } : {\alpha \in I } \}$ into a group $G$ is a one-to-one mapping of the union $\cup _ {\alpha \in I } G _ \alpha$ into $G$ whose restriction to each $G _ \alpha$ is an isomorphism. An amalgam of groups in which all intersections $G _ \alpha \cap G _ \beta$ are identical (and equal to, say, a subgroup $H$) is imbeddable in the group that is the free product of the groups $G _ \alpha$ with the amalgamated subgroup $H$. 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 $\sigma , \tau$ 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 $\sigma$ is imbeddable in a group with the property $\tau$. 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 $\sigma , \tau$ denote solvability, nilpotency, periodicity, local finiteness, etc. (in different combinations).

In the definition of amalgam above, think of the $G _ \alpha$ as all being subsets of some large set $X$. The amalgamated product of groups $G _ {i}$" over a common subgroup U" is constructed as follows. Let $G _ {i}$ be a set of groups indexed by the set $I$. For each $i$ let $U _ {i}$ be a subgroup of $G _ {i}$ and for each $i$ let there be an isomorphism $\phi _ {i} : U _ {i} \rightarrow U$ identifying $U _ {i}$ and $U$. Consider the set $\widetilde{G}$ of all words

$$a _ {1} \dots a _ {t}$$

with each $a _ {i}$ from some $G _ {j}$, and consider the following elementary equivalences

1) if $a _ {i} = 1$ then $a _ {1} \dots a _ {i-1} a _ {i} a _ {i+1} {} \dots a _ {t}$ is equivalent to $a _ {1} \dots a _ {i-1} a _ {i+1} \dots a _ {t}$;

2) if $a _ {i}$ and $a _ {i+1}$ belong to the same group $G _ {j}$ and $a _ {i} a _ {i+1} = a _ {i} ^ \prime$ in $G _ {j}$ then $a _ {1} \dots a _ {i-1} a _ {i} a _ {i+1} \dots a _ {t}$ is equivalent to $a _ {1} \dots a _ {i-1} a _ {i} ^ \prime a _ {i+2} \dots a _ {t}$;

3) if $a _ {i} = u _ {i} \in U _ {j} \subseteq G _ {j}$ and $b _ {i} = u _ {k} \in U _ {k} \subseteq G _ {k}$ and $\phi _ {k} ( u _ {k} ) = u = \phi _ {i} ( u _ {i} ) \in U$, then $a _ {1} \dots a _ {i-1} a _ {i} a _ {i+1} \dots a _ {t}$ is equivalent to $a _ {1} \dots a _ {i-1} b _ {i} a _ {i+1} \dots a _ {t}$.

Let $\sim$ be the equivalence relation generated by these elementary equivalences, then $\widetilde{G} / \sim$ is the amalgamated product of the $G _ {i}$, more precisely of the $( G _ {i} , U _ {i} )$( i.e. the free product with amalgamated subgroup of the $G _ {i}$ $U$); the group law is induced by concatenation.

Amalgamated products are non-trivial. This follows from the following canonical form theorem. For each $i$ select a set $R _ {i}$ of left coset representatives of $U$ in $G _ {i}$. Then each word is equivalent to precisely one of the form $uz _ {1} \dots z _ {t}$ with each $z _ {i}$ in some $R _ {j}$, $u \in U$ and $z _ {1}$ and $z _ {i+1}$ belonging to different $G _ {j}$' s for $i = 1 \dots t-1$. If $U = \{ e \}$ one obtains of course the free product of the $G _ {i}$. 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 $U$ is the amalgamated subgroup, then one can take subgroups $H _ {i}$ of the $G _ {i}$ with different intersections with $U$ so that the $H _ {i}$ 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.

How to Cite This Entry:
Amalgam of groups. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Amalgam_of_groups&oldid=45095
This article was adapted from an original article by Yu.I. MerzlyakovN.S. Romanovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article