# Are Fundamental Theorem of Homomorphism and First theorem of Isomorphism the same?

In many instances, the fundamental theorem of homomorphism and the first theorem of isomorphism are considered the same. In a book I encountered the following different statements.

Fundamental Theorem of Homomorphism. Let $$G$$ be a group. If $$N$$ is a normal subgroup $$G$$, then $$\frac{G}{N}$$ is a homomorphic image of $$G$$. Conversely, if any group $$G'$$ is a homomorphic image of $$G$$ then $$G'$$ is isomorphic to some quotient group of $$G$$. If fact, if $$f$$ is a homomorphism of $$G$$ onto $$G'$$, then $$G'$$ is isomorphic to $$\frac{G}{\ker f}$$.

First Theorem of Isomorphism. Let $$f$$ be a homomorphism of a group $$G$$ onto $$G'$$ and $$H=\ker f$$, $$K'$$ any normal subgroup of $$G'$$ and $$K=\lbrace x\in G| f(x)\in K' \rbrace=f^{-1}(K').$$ Then $$K$$ is a normal subgroup of $$G$$ containing $$H$$ and $$\frac{G}{K}$$ is isomorphic to $$\frac{G'}{K'}$$.

In my opinion, the two statements are different. Please enlighten me whether

Fundamental Theorem of Homomorphism and First Theorem of Isomorphism are same or different?

• The second theorem implies the first one easily. The first implies the second with a little work. What each one is called may vary from text to text. Commented Sep 28, 2021 at 14:59
• @EthanBolker well. So, could I tell my juniors that the two theorems are same or they are different? Because in youtubes, they are saying that the two theorems mean the same.
– gete
Commented Sep 28, 2021 at 15:24