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''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041620/f0416201.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041620/f0416202.png" />''
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$#C+1 = 29 : ~/encyclopedia/old_files/data/F041/F.0401620 Free product of groups
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A group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041620/f0416203.png" /> generated by the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041620/f0416204.png" /> such that any homomorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041620/f0416205.png" /> of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041620/f0416206.png" /> into an arbitrary group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041620/f0416207.png" /> can be extended to a homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041620/f0416208.png" />. The symbol * is used to denote a free product, for example,
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<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/f/f041/f041620/f0416209.png" /></td> </tr></table>
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'' $  G _ {i} $,
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$  i \in I $''
  
in the case of a finite set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041620/f04162010.png" />. Each element of a free product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041620/f04162011.png" /> that is not the identity can be expressed uniquely as an irreducible word <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041620/f04162012.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041620/f04162013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041620/f04162014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041620/f04162015.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041620/f04162016.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041620/f04162017.png" /> is defined by sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041620/f04162018.png" /> of generators and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041620/f04162019.png" /> of defining relations, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041620/f04162020.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041620/f04162021.png" />. Then the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041620/f04162022.png" /> defined by the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041620/f04162023.png" /> of generators and the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041620/f04162024.png" /> of defining relations is the free product of the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041620/f04162025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041620/f04162026.png" />.
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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,
  
Every subgroup of a free product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041620/f04162027.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041620/f04162028.png" /> in the free decomposition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041620/f04162029.png" /> (Kurosh' theorem).
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$$
 +
G  = \
 +
\prod _ {i \in I } {}  ^ {*} G _ {i} ,\  \textrm{ and } \ \
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G  =  G _ {1} * \dots * G _ {k}  $$
 +
 
 +
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====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.G. Kurosh,  "The theory of groups" , '''1–2''' , Chelsea  (1955–1956)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  W. Magnus,  A. Karrass,  B. Solitar,  "Combinatorial group theory: presentations in terms of generators and relations" , Wiley (Interscience)  (1966)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.G. Kurosh,  "The theory of groups" , '''1–2''' , Chelsea  (1955–1956)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  W. Magnus,  A. Karrass,  B. Solitar,  "Combinatorial group theory: presentations in terms of generators and relations" , Wiley (Interscience)  (1966)</TD></TR></table>
 
 
  
 
====Comments====
 
====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|Amalgam of groups]]).
 
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|Amalgam of groups]]).

Latest revision as of 19:40, 5 June 2020


$ 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).

How to Cite This Entry:
Free product of groups. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Free_product_of_groups&oldid=18482
This article was adapted from an original article by A.L. Shmel'kin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article