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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?

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  • $\begingroup$ 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. $\endgroup$ Sep 28 '21 at 14:59
  • $\begingroup$ @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. $\endgroup$
    – gete
    Sep 28 '21 at 15:24
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Comment rewritten as an answer in response to comment on the question.

I think that asking the simple yes/no question as to whether these two statements about groups are "the same theorem" or "different theorems" is not useful.

Formally, two theorems are "the same" just when they have the same hypotheses and same conclusions. But that can be subtle. If you have two theorems about groups but each starts from a different (but logically equivalent) definition of a group are they the same?

I think it's better to tell your students that the names given to mathematical theorems can vary. Sometimes a theorem will have several names. Sometimes different theorems will have the same name. Often in those cases each can be derived relatively easily from the other - but the arguments though easy in both directions may be somewhat more difficult in one of them. In this case the second theorem implies the first one easily. The first implies the second with a little work.

In any particular context (a paper in the literature, a textbook, a course) the particular theorem referred to as "The Fundamental Theorem on Isomorphisms" or "The Fundamental Theorem of Calculus" should be precisely specified.

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