a subgroup of index 2 is normal
Let be a group and let be a subgroup of of index 2. Then is normal in .
Let be a group and let be an index 2 subgroup of . By definition of index, there are only two left cosets of in , namely:
where is any element of which is not in . Notice that if are two elements in which are not in then belongs to . Indeed, the coset (because would immediately yield ) and so and .
Let be an arbitrary element of and let . If then and we are done. Otherwise, assume that . Thus and by the remark above , as desired. ∎
|Title||a subgroup of index 2 is normal|
|Date of creation||2013-03-22 15:09:25|
|Last modified on||2013-03-22 15:09:25|
|Last modified by||alozano (2414)|