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free product with amalgamated subgroup
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(Definition)
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Definition 1 Let  ,  be groups and
 ,  be monomorphisms. The free product of  and  with amalgamated subgroup  , is defined to be a group  that has the following
two properties
- there are homomorphisms
 ,  that make the following diagram commute
 is universal with respect to the previous property, that is for any other group  and homomorphisms
 ,  that fit in such a commutative diagram there is a unique homomorphism  so that the following diagram commutes
It follows by “general nonsense” that the free product of  and  with amalgamated subgroup  , if it exists, is “unique up to unique isomorphism.” The free product of  and  with amalgamated subgroup  , is denoted by
 . The following theorem asserts its existence.
Theorem 2
 exists for any groups  ,  and monomorphisms
 ,  .
Notice that in the above proof it would be sufficient to divide by the relations
 for  in a generating set of  . This is useful in practice when one is interested in obtaining a presentation of
 .
In case that the  's are not injective the above still goes through verbatim. The group thusly obtained is called a “pushout”.
Examples of free products with amalgamated subgroups are provided by Van Kampen's theorem.
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"free product with amalgamated subgroup" is owned by mathcam. [ full author list (2) | owner history (2) ]
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Cross-references: Van Kampen's theorem, injective, generating set, relations, divide, sufficient, quotient group, universal properties, canonical projection, normal closure, generators, word, presentation, inclusion, without loss of generality, proof, commutative diagram, universal, diagram, homomorphisms, properties, subgroup, monomorphisms, groups
There are 3 references to this entry.
This is version 4 of free product with amalgamated subgroup, born on 2003-01-30, modified 2005-01-28.
Object id is 3944, canonical name is FreeProductWithAmalgamatedSubgroup.
Accessed 8766 times total.
Classification:
| AMS MSC: | 20E06 (Group theory and generalizations :: Structure and classification of infinite or finite groups :: Free products, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations) |
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Pending Errata and Addenda
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