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)

Comments

The notion of a free product of groups is a special case of that of a free product with amalgamated subgroup (see Amalgam of groups).