simple algebraic system
For example, let’s find out what are the simple algebras in the class of groups. Let be a group that is simple in the sense defined above.
First, what are the congruences on ? A congruence on is a subgroup of and an equivalence relation on at the same time. As an equivalence relation, corresponds to a partition of in the following manner: and , where for . Each of the is an equivalence class of . Let be the equivalence class containing . If , then , so that , or . In addition, , so . is a subgroup of . Furthermore, if , , so that , is a normal subgroup of . Conversely, given a normal subgroup of , forming left (right) cosets of , and taking gives us the congruence on .
Now, if is simple, then this says that the only congruences on are and , which corresponds to having and as the only normal subgroups. So, as a simple algebra is just a simple group. Conversely, if is a simple group, then the only congruences on are those corresponding to and , the only normal subgroups of . Therefore, a simple group is a simple algebra.
Remark. Any simple algebraic system is subdirectly irreducible.
|Title||simple algebraic system|
|Date of creation||2013-03-22 16:46:56|
|Last modified on||2013-03-22 16:46:56|
|Last modified by||CWoo (3771)|