Normal iff potential endomorphism kernel
This article gives a proof/explanation of the equivalence of multiple definitions for the term normal subgroup
View a complete list of pages giving proofs of equivalence of definitions
- Endomorphism kernel implies normal
- Normality satisfies intermediate subgroup condition
(2) implies (1) (the easy direction)
If (2) holds, is a normal subgroup of by Fact (1). By Fact (2), then, is also a normal subgroup of .
(1) implies (2) (the hard direction)
Identify with the first direct factor (i.e., treating it as an internal direct product locally) and with the subgroup in the first direct factor.
By construction is the kernel of the endomorphism of that sends each direct factor to the next, with the first map being the quotient map by , and the remaining maps being identity maps. Explicitly, if is the quotient map, the mapping we are talking about is:
(where denotes the identity element).