# 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.

#### References

 [a1] M. Hall jr., "The theory of groups" , Macmillan (1959) [a2] H. Neumann, "Generalized free products with amalgamated subgroups I" Amer. J. Math. , 70 (1948) pp. 590–625 [a3] H. Neumann, "Generalized free products with amalgamated subgroups II" Amer. J. Math. , 71 (1949) pp. 491–540
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