In a certain book the following definition is given: Let G be a group. A normal subgroup N 1 of G is a minimal normal subgroup of G if 1 and N are the only normal subgroups of G that are contained in N.
Then the following proposition is stated: Let be a minimal normal subgroup of . If is an epimorphism from to a group , then or is a minimal normal subgroup of .
The author gives this proof: Let be a normal subgroup of that is contained in . Then is a normal subgroup of , and since . Hence and .
OK. But why ? Since is homomorphism and normal in , is normal in . And the intersection of two normal subgroups is a normal subgroup, that is is normal in . But and N is minimal normal in . Hence either or . Hence, if it were true that , we would have . Assuming the latter is true, we have the following: , . But since is onto, . Hence . But by assumption. Hence .
As you see, I can follow the steps in the proof, except for the inference " ". Any suggestion?
The book is The Theory of Finite Groups, An Introduction, by Kurzweil and Stellmacher, and the proposition is proposition 1.7.1, p. 36.