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Anti—Representations
| Content Provider | Scilit |
|---|---|
| Author | Mader, A. |
| Copyright Year | 2000 |
| Description | An almost completely decomposable group X is a finite essential extension of a completely decomposable group A. Hence there is an integer e such that eX C A, and one obtains a simplified image of X in the finite group A /e A , namely eX /eA . Are there any essential properties of X that can be studied by looking at the fi nite group A = A /eA together with its subgroup eX = e X /e A l The answer is no and yes. The group A has much less structure than the completely decompos able A. It is easily possible that A is a free Z/eZ-module with many endomorphisms and automorphisms while A has a very rigid structure with endomorphisms and automorphisms severely restricted by the fully invariant type subgroups A(r). Considering A and its subgroup eX is not good enough. But the approximation of (A ,e X ) to (A, AT) is closer if A is furnished with the distinguished subgroups A(t) together with eX . By looking at A with these distinguished subgroups it is possible to decide whether A is regulating in X (Proposition 8.1.9) or the regulator of X (Theorem 8.1.10), one can decide whether two almost completely decompos able groups X and Y with common regulator are isomorphic (Theorem 8.1.10), and one can see that there is a natural weakening of isomorphism, called typeisomorphism (Definition 8.1.14). Classification up to type-isomorphism becomes the study of finite abelian groups with distinguished subgroups. Type-isomorphism can be compared with isomorphism and it will become clear why and to what ex tend type-isomorphism simplifies the classification problem. It remains to be seen in later chapters that type-isomorphism is fine enough to preserve important prop erties. Book Name: Almost Completely Decomposable Groups |
| Related Links | https://content.taylorfrancis.com/books/download?dac=C2009-0-18781-0&isbn=9780429078439&doi=10.1201/9781482287486-14&format=pdf |
| Ending Page | 180 |
| Page Count | 16 |
| Starting Page | 165 |
| DOI | 10.1201/9781482287486-14 |
| Language | English |
| Publisher | Informa UK Limited |
| Publisher Date | 2000-03-09 |
| Access Restriction | Open |
| Subject Keyword | Book Name: Almost Completely Decomposable Groups Classification Decomposable Distinguished Subgroups |
| Content Type | Text |
| Resource Type | Chapter |
