# Alternate definition of Group Kernel

In my Algebra class, my professor defines the kernel in the context of group homomorphisms, but this seems completely unrelated to the normal definition. Is these equivalent to the normal definition? It looks completely different.

Let $G, H$ be groups and $u, v: G \to H$ be homomorphisms of groups. We define the pair $(K, i)$ to be the kernel of the pair $(u, v)$ provided $i: K \to G$ is a homomorphism such that $u \circ i = v \circ i$ and, for any group $F$ and homomorphism $w : F \to G$ such that $u \circ w = v \circ w$, there is a unique homomorphism $w' : F \to K$ such that $w = i \circ w'$.

EDIT: One commenter (now deleted) suggested that this is the category theoretic definition of the kernel. It looks to me that the definition given on Wikipedia is still slightly different than the definition given to me in class. The Wikipedia definition defines the kernel of a some morphism $f$, while my professor defines the kernel of a pair $(u, v)$.

• I still don't completely understand. My professor defined the kernel in terms of a pair of (homo)morphisms (u,v), while the category theoretic definition speaks of the kernel of (just one) morphism f. – Nitin Sep 27 '15 at 18:49

The kernel, as defined in your question, is what is nowadays usually called the equalizer of $u$ and $v$. At one time, the terminology "kernel" or "difference kernel" was fairly common, but I don't think it's used very much in this sense any more. Nowadays, at least in group theory (and ring theory), one speaks of the kernel of a single homomorphism $f$. This is a special case of equalizers; the kernel of $f$ is the equalizer of $f$ and the "zero" homomorphism (meaning the homomorphism that sends all elements of the domain group to the identity element of the codomain group). More precisely, the kernel of $f:G\to H$ is the subgroup $K$ of $G$ consisting of the elements of $G$ that $f$ maps to the identity of $H$. The inclusion homomorphism $K\to G$ (sending every element of $K$ to itself considered as an element of $G$) is the equalizer of $f$ and the homomorphism that sends all of $G$ to the identity element of $H$.