The amalgamated (free) product of groups after the group![G_ {i}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0dd9fe8d455762608cc4e0a946b452492790ee5f)
or the free product of the groups with the
amalgamated subgroup is a mathematical construction related to the free product of groups. First, the free product of the groups is formed. However, these must all contain a subgroup that is isomorphic to the (sub) group . Then these subgroups are merged (figuratively speaking) and amalgamated in this sense by (a) suitable identification of elements and (b) adaptation of the group linkage within the free product . The identification of two elements from different of the predefined subgroups that are too isomorphic is achieved via the isomorphism to the group (see below Ä3 ) and the group linkage is adjusted accordingly (see below group linkage ).
![U](https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025)
![U](https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025)
![U](https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025)
![U](https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025)
One speaks of a nontrivial amalgamated product if and is.
![{\ displaystyle U \ not = G_ {1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ba22d3a4dc194f3513d8c7a97ab85acb93f0888c)
![{\ displaystyle U \ not = G_ {2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6a33f0aa4bdeeb6d372e3bff3b071c89b1e30734)
The amalgamated product of two groups and with a common subgroup is an example of a pushout .
![G_ {1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e6ea4f4668b8334c8a7d3d284b0fd22131ef5f52)
![G_ {2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/645011b0c6933a02f5f7d84624f78220d747427e)
![U](https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025)
The free product is an application or a special case of the amalgamated product, since every free product can be regarded as an amalgamated product by virtue of the amalgamation according to the trivial subgroup of its factors.
![\{1\}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5acdcac635f883f8b4f0a01aa03b16b22f23b124)
Definition (constructive)
basic requirements
Let be an index set and a family of groups . Furthermore, each of these groups contains a subgroup , and all of these are isomorphic to a group. The associated group isomorphism , which mediates this isomorphism, is denoted by for all .
![I {\ stackrel {{\ mathrm {def}}} =} \ {1,2, \ ldots, n \}, n \ in \ mathbb {N}](https://wikimedia.org/api/rest_v1/media/math/render/svg/125a5965b8753b54d3d40b3e19be6b776e11dc16)
![U_ {i}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b21a6f475b0e68475c6019abe1fed0b415e0e42)
![U_ {i}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b21a6f475b0e68475c6019abe1fed0b415e0e42)
![U.](https://wikimedia.org/api/rest_v1/media/math/render/svg/7a305ef479ab152035f334467a2c314baa23eb36)
![\ varphi _ {i} \ colon U_ {i} {\ xrightarrow {\ cong}} U](https://wikimedia.org/api/rest_v1/media/math/render/svg/cff47006b8c89c252e1145e1beccfea2918aad1e)
![i \ in I.](https://wikimedia.org/api/rest_v1/media/math/render/svg/2d740fe587228ce31b71c9628e089d1a9b37c6be)
Be further - a word about that is written one after the other (stringing, chaining, etc.)![t \ in \ mathbb {N}.](https://wikimedia.org/api/rest_v1/media/math/render/svg/1ed233e109e38a8776d480031d8bb575eff761d4)
![a_ {1} a_ {2} \ ldots a_ {t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7d7230279d7a5890c3b46e3b6bf8789c401784ff)
of elements from being understood that either (for ) empty was - then written or - or there is for each one so that (two different. e. applies need not be from the same group).
![G_ {i},](https://wikimedia.org/api/rest_v1/media/math/render/svg/ab444d13150edd05eb02dfc8c504610215f24e79)
![t = 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/43469ec032d858feae5aa87029e22eaaf0109e9c)
![1](https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf)
![\epsilon](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3837cad72483d97bcdde49c85d3b7b859fb3fd2)
![i = 1, \ ldots, t](https://wikimedia.org/api/rest_v1/media/math/render/svg/8bbd4d1c9ec10ab74861a7471734448ca1d65f3b)
![j_ {i} \ in I,](https://wikimedia.org/api/rest_v1/media/math/render/svg/9fc7b4caacd504becce8c2efd5cb1a00c0df0dfd)
![a_ {i} \ in G _ {{j_ {i}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ba30824da666e53b47d051db60407459542d21a0)
![a_ {i}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0bc77764b2e74e64a63341054fa90f3e07db275f)
In the following we write for the empty word as well as for the neutral elements of the groups without distinction .
![{\ displaystyle 1_ {j}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/19275c8cc1e2f88940978a78e259e6c518073966)
![G_ {j}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3017ab5e70c54326ca950dc1f46f61534c44a54a)
![1](https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf)
Equivalence relation
Elementary equivalences
Analogous to the procedure for the formation of the free product of the groups , we now consider words from elements from the and define so-called elementary equivalences ((1 - –3) between them:
![G_ {i}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0dd9fe8d455762608cc4e0a946b452492790ee5f)
![G_ {i}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0dd9fe8d455762608cc4e0a946b452492790ee5f)
(Ä1)
- - Neutral elements can be omitted. -
- If so then
![a_ {i} = 1,](https://wikimedia.org/api/rest_v1/media/math/render/svg/bdaabcd941fb201014629035aaf2adbd80e1e25f)
![a_ {1} \ ldots a _ {{i-1}} {\ color {RawSienna} a_ {i}} a _ {{i + 1}} \ ldots a_ {t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4c796631c0eea5a3799fb28acb59fd181f8d1f58)
-
(elementary) equivalent to
(Ä2)
- - Two elements can be replaced by your product. -
- If and are from the same group and in holds, then be
![a_ {i}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0bc77764b2e74e64a63341054fa90f3e07db275f)
![a _ {{i + 1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/714e1a37379d8e664890d0385b0bd842df2ad6e5)
![G_ {j}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3017ab5e70c54326ca950dc1f46f61534c44a54a)
![a_ {i} a _ {{i + 1}} = a ^ {*}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5d7bce02c36a93c4136695133a060f59699313c1)
![G_ {j}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3017ab5e70c54326ca950dc1f46f61534c44a54a)
![a_ {1} \ ldots {\ color {RawSienna} a_ {i} a _ {{i + 1}}} \ ldots a_ {t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d26c3273c422835e5bf3ddfd4ef3491472d7e816)
-
(elementary) equivalent to
(Ä3)
- Intermediate remark:
- We say that two elements and with are each associated , if, by means of the isomorphisms between and the same match; d. H. if applicable.
![u_ {j} \ in U_ {j} \ subseteq G_ {j}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c6641b31cf406446eec4a28a8b1ae0a60bf24eff) ![u_ {k} \ in U_ {k} \ subseteq G_ {k}](https://wikimedia.org/api/rest_v1/media/math/render/svg/99d0f9c85601e6e5406c7d3867108a4cb11e8599) ![j, k \ in I.](https://wikimedia.org/api/rest_v1/media/math/render/svg/e19a406aa063514b7ac100e747a74c998e8b0d34) ![\ varphi _ {j}, \ \ varphi _ {k}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d2c59c5f15728fef09529175f3441080f4e1e1b6) ![U_ {k}](https://wikimedia.org/api/rest_v1/media/math/render/svg/924cf85ae48192ae1c1ce15c109032e0a69b1014) ![U,](https://wikimedia.org/api/rest_v1/media/math/render/svg/59c21c569e498e0197798e3428b2bc6f25c0a457) ![u \ in U](https://wikimedia.org/api/rest_v1/media/math/render/svg/65c3e0203fca1c3b6c1982520132b9f057ed5d88) ![\ varphi _ {k} (u_ {k}) = u = \ varphi _ {j} (u_ {j})](https://wikimedia.org/api/rest_v1/media/math/render/svg/6c4683b6f04d06bd186a7687ce535ea3667bacbf)
|
- - Elements can be replaced by associated elements. -
- If and with and the elements and belong to each other, then be
![a_ {i} = u_ {j} \ in U_ {j} \ subseteq G_ {j}](https://wikimedia.org/api/rest_v1/media/math/render/svg/657f05ad847b92596017c8c27f36ea0a5a3563ca)
![b_ {i} = u_ {k} \ in U_ {k} \ subseteq G_ {k}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aa7ec36ab49d53cf3cea9719158f255568859503)
![j, k \ in I.](https://wikimedia.org/api/rest_v1/media/math/render/svg/e19a406aa063514b7ac100e747a74c998e8b0d34)
![u_ {j}](https://wikimedia.org/api/rest_v1/media/math/render/svg/86f8d397f4684f948df846413e6d62b009718724)
![u_ {k}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9837644700489d04d977da272524cd5fda36f3d7)
![a_ {1} \ ldots a _ {{i-1}} {\ color {RawSienna} a_ {i}} a _ {{i + 1}} \ ldots a_ {t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4c796631c0eea5a3799fb28acb59fd181f8d1f58)
-
(elementary) equivalent to
Word-by-word equivalence
On the basis of the elementary equivalences E1 – E3 we now explain the word-wise equivalence : Two words and are (word-wise) equivalent if there is a sequence
![x](https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4)
![y](https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d)
![x {\ stackrel {{\ mathrm {def}}} =} x_ {1}, \ x_ {2}, \ \ ldots, \ x_ {m} {\ stackrel {{\ mathrm {def}}} =} y](https://wikimedia.org/api/rest_v1/media/math/render/svg/7e570e5c5ce6038b2c6b0dc10d1ba4830545fcc0)
with gives, in which and for each are elementarily equivalent. (The word-wise equivalence thus corresponds to the transitive envelope or the reflexive-transitive envelope of the elementary equivalence.) For the equivalence relation defined in this way, we use the symbol for (word-wise or elementary) equivalent words in the following and thus applies![m \ in \ mathbb {N}](https://wikimedia.org/api/rest_v1/media/math/render/svg/42411e85d874a733209223302bbd8d5e3ad04cb0)
![x_ {i}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e87000dd6142b81d041896a30fe58f0c3acb2158)
![x_ {i + 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bc56fe176df2317239339ad413c58d09cd1c187b)
![i = 1, \ ldots, m-1](https://wikimedia.org/api/rest_v1/media/math/render/svg/560fc9a915c8235951b486895eae7cd31cd6398b)
![\ sim.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4580b0650bad5e143167d934da1af672cc273393)
![x](https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4)
![y](https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d)
Let us denote by the set of all words that mean so
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b)
![G_ {i},](https://wikimedia.org/api/rest_v1/media/math/render/svg/ab444d13150edd05eb02dfc8c504610215f24e79)
![[x] = \ {y \ in G | y \ sim x \}](https://wikimedia.org/api/rest_v1/media/math/render/svg/855478f012ee42f3bf32af9d6ac6a94fe77d3243)
the set of all equivalent words. This is also called the equivalence class of .
![x](https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4)
![x](https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4)
Group link
On top of the set is a combination of words and a link
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b)
![x = a_ {1} \ ldots a_ {t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e21930274d61d64a73d0eb212d1a1b52f3aebf41)
![y = b_ {1} \ ldots b_ {s}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8787e3a229e4b26693ff6240e0529ea98a906845)
![xy = a_ {1} \ ldots a_ {t} b_ {1} \ ldots b_ {s}](https://wikimedia.org/api/rest_v1/media/math/render/svg/426b2ec7f12ac6e923bb1df76add9c30505cf111)
explained. We finally transfer this connection in a natural way to the quotient set from according to the equivalence relation by defining:
![G / {\ sim}](https://wikimedia.org/api/rest_v1/media/math/render/svg/02d52a55182b03ede6fe34721c2b497e1aab36fe)
![\ sim,](https://wikimedia.org/api/rest_v1/media/math/render/svg/8b5e05b1a2f9111e6902951193d6af88b0886558)
-
![[x] \ ast [y] {\ stackrel {{\ mathrm {def}}} =} [xy].](https://wikimedia.org/api/rest_v1/media/math/render/svg/49d6694ee0c40e08854b2581bc9efa54ece7c0a3)
The product (in G / ~) of the equivalence classes of x and y is equal to the equivalence class of the product (in G) of x and y.
This product is also called the canonical product on designated.
![G / {\ sim}](https://wikimedia.org/api/rest_v1/media/math/render/svg/02d52a55182b03ede6fe34721c2b497e1aab36fe)
Final definition
The quotient set together with the product just defined forms a group, namely the amalgamated product of the groups or the free product of the groups with the amalgamated subgroup![\ left (G / {\ sim}, \ ast \ right),](https://wikimedia.org/api/rest_v1/media/math/render/svg/2e21ff9b9569563cbdd520cd6d57db1f14005118)
![G_ {i}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0dd9fe8d455762608cc4e0a946b452492790ee5f)
![G_ {i}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0dd9fe8d455762608cc4e0a946b452492790ee5f)
Web links
literature
- Hall, Marshall: The theory of groups. Macmillan, New York, 1959.
Footnotes / individual references
-
↑ See the corresponding Wiktionary entry under web links .
-
↑ The subgroups are trivially all isomorphic to any group of group order 1. Choose the amalgamation subgroup .
![\{1\}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5acdcac635f883f8b4f0a01aa03b16b22f23b124)
-
↑ The result is where and are set. - "When interpreting the formula, equal signs are evaluated before commas."
![{\ displaystyle x_ {1}, x_ {2}, \ ldots, x_ {m}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8e347ad0ab6f4b8fde8571341761c9567a9b6063)
![{\ displaystyle x_ {1} {\ stackrel {\ mathrm {def}} {=}} x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7738b374097aabd12c3759eccdde36d43828958b)
-
↑ "canonical" means something like that the product in the canon of mathematics, i. For example, it is usually defined in this way in the leading mathematics literature and can be regarded as an exemplary, binding and / or reliable convention in this sense. The term comes from canon law and its designation as canon law .