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Groups - Basic Properties
| Content Provider | Scilit |
|---|---|
| Author | Effinger, Gove Mullen, Gary L. |
| Copyright Year | 2019 |
| Description | Having learned the formal definition of a group and seen a number of examples of different kinds of groups in the previous chapter, we shall start this chapter by summarizing some of the basic properties which all groups share. We wish to emphasize again that when working with an abstract group G, we usually suppress the * notation and simply use juxtaposition (i.e., writing ab rather that a * b), which means we will refer to the operation as “multiplication.” Remember, however, that in any given example of a group, the operation may well not be multiplication, but may instead be addition of numbers, addition of matrices, composition of permutations, rigid motions of an object, and so on. Theorem 22.1 (Basic Group Properties) The identity element e of every group G is unique. For each element a in a group G, the inverse a$ ^{−1}$ of a is unique. For all a, b in a group G, each of the equations ax = b and xa = b has a unique solution x in G. For all a and b in a group G, $(ab)^{−1}$ = b$ ^{−1}$ a$ ^{−1}$. We prove (ii) below and note that you were asked to establish all four of these properties in Exercises 21.5, 21.6, 21.7 and 21.9. If you did not do those exercises previously, do them now, making use of the hints and suggested solutions given there if you need them. You should note that in proving the latter three properties in this theorem, you do need to make use of the fact that the binary operation is associative. 154Proof of (ii). Suppose that b and c are both inverses of an element a in G, whose identity is denoted by e. Then using the definitions of identity and inverse and using the associativity of the operation, we have b = b e = b ( a c ) = ( b a ) c = e c = c . We conclude that inverses are unique. Book Name: An Elementary Transition to Abstract Mathematics |
| Related Links | https://content.taylorfrancis.com/books/download?dac=C2019-0-00596-8&isbn=9780429324819&doi=10.1201/9780429324819-22&format=pdf |
| Ending Page | 158 |
| Page Count | 6 |
| Starting Page | 153 |
| DOI | 10.1201/9780429324819-22 |
| Language | English |
| Publisher | Informa UK Limited |
| Publisher Date | 2019-11-05 |
| Access Restriction | Open |
| Subject Keyword | Book Name: An Elementary Transition To Abstract Mathematics History and Philosophy of Science Properties Definition Operation Exercises Examples Theorem |
| Content Type | Text |
| Resource Type | Chapter |
