# Free product of groups

$G _ {i}$, $i \in I$

A group $G$ generated by the groups $G _ {i}$ such that any homomorphisms $\phi _ {i} : G _ {i} \rightarrow H$ of the $G _ {i}$ into an arbitrary group $H$ can be extended to a homomorphism $\phi : G \rightarrow H$. The symbol * is used to denote a free product, for example,

$$G = \ \prod _ {i \in I } {} ^ {*} G _ {i} ,\ \textrm{ and } \ \ G = G _ {1} * \dots * G _ {k}$$

in the case of a finite set $I$. Each element of a free product $G$ that is not the identity can be expressed uniquely as an irreducible word $v = g _ {i _ {1} } \dots g _ {i _ {n} }$, where $g _ {i _ {j} } \in G _ {i _ {j} }$, $g _ {i _ {j} } \neq 1$ and $i _ {j} \neq i _ {j + 1 }$ for any $j = 1 \dots n - 1$. The construction of a free product is important in the study of groups defined by a set of generating elements and defining relations. In these terms it can be defined as follows. Suppose that each group $G _ {i}$ is defined by sets $X _ {i}$ of generators and $\Phi _ {i}$ of defining relations, where $X _ {i} \cap X _ {j} = \emptyset$ if $i \neq j$. Then the group $G$ defined by the set $X _ {i} = \cup _ {i \in I } X _ {i}$ of generators and the set $\Phi = \cup _ {i \in I } \Phi _ {i}$ of defining relations is the free product of the groups $G _ {i}$, $i \in I$.

Every subgroup of a free product $G$ can be decomposed into a free product of subgroups, of which some are infinite cyclic and each of the others is conjugate with some subgroup of a group $G _ {i}$ in the free decomposition of $G$( Kurosh' theorem).

#### References

 [1] A.G. Kurosh, "The theory of groups" , 1–2 , Chelsea (1955–1956) (Translated from Russian) [2] W. Magnus, A. Karrass, B. Solitar, "Combinatorial group theory: presentations in terms of generators and relations" , Wiley (Interscience) (1966)