proof of second isomorphism theorem for groups
First, we shall prove that is a subgroup of : Since and , clearly . Take . Clearly . Further,
Also, for , , since
and since is a normal subgroup of . So is closed under inverses, and is thus a subgroup of .
Since is a subgroup of , the normality of in follows immediately from the normality of in .
Clearly is a subgroup of , since it is the intersection of two subgroups of .
and if , then we must have . So
Thus, since and , by the First Isomorphism Theorem we see that is normal in and that there is a canonical isomorphism between and .
|Title||proof of second isomorphism theorem for groups|
|Date of creation||2013-03-22 12:49:47|
|Last modified on||2013-03-22 12:49:47|
|Last modified by||yark (2760)|