Why is the kernel of a group homomorphism so called? I took a course in Linear Algebra last fall. I have come to associate the word kernel with the nullspace of matrices/linear transformations - and now that I am studying Group Theory, similar intuition doesn't seem to carry forward.
Why would anyone name it kernel (in group theory) if it really isn't the same thing as that in linear algebra? Perhaps there is a relation or connect between the two definitions that I am missing. Could someone help me better understand why the kernel of a homomorphism is so defined?
To document the definitions:

(Linear Algebra): Consider a linear transformation $T:V\to W$. $$\ker T = \{x\in V: T(x) = 0\}$$
(Group Theory): Let $\phi:G\to H$ be a group homomorphism. $$\ker\phi = \{x\in G: \phi(x) = 1_H\}$$ where $1_H$ is the identity in $H$.

Thanks!
 A: It might be worth noting that it appears that the use of "kernel" in group theory precedes the use in Linear Algebra. According to the Earliest known uses of some of the words in mathematics website, though the term "kernel" was already in use in Fourier analysis and the theory of integral equations, it first appeared in the algebraic context in Pontrjagin's 1931 paper, Über den algebraischen Inhalt topologischer Dualitätssätze in Math. Annalen 105. The quotation, translated, reads "The set of all the elements of the group $G$ which go into the identity of the group $G^*$ under the homomorphism $g$ is called the kernel of this homomorphism." The use in linear algebra came from Birkohff and Mac Lane's "Survey of Modern Algebra", 3rd edition, where they write:

Since $\mathbf{0}$ is the group identity, it follows that that the null-space of $T$ is precisely the kernel of $T$ regarded as a group-homomorphism.

"Kernel" can either mean "the inner, softer part of a seed, fruit stone, or nut", or "the central or essential part". I suspect, but this is mere surmise, that the use in algebra is related to the first meaning, thinking about the things that get "squished" into the trivial element, at the "center" or "middle" of the group.
A: The kernel of a linear map and of a group homomorphism is just the subset of the domain consisting of all elements which are mapped to the trivial element in the codomain. This encapsulates their general role.
When given a linear map $T\colon V\to W$ or a group homomorphism $\phi\colon G\to H$ we have to make precise what the trivial element in the codomain is. In case of vector spaces it is the zero vector and in case of groups it is the identity element, here $0_W$ and $1_H$, respectively. However, a vector space is in the end just an (abelian) group where the zero vector is the identity element. Hence, the two notions conincide in this case (note that scalar multiplication is not relevant here as $\varphi(av)=a\varphi(v)$ for every $a$ in the base field).
So, viewing $V$ and $W$ as additive groups $(G,+,1_G)=(V,+,0_V)$ and $(H,+,1_H)=(W,+,0_W)$ the group-theoretic kernel of $\phi\colon G\to H\triangleq T\colon V\to W$ becomes the null space:
$$\ker\phi=\{x\in G\,|\,\phi(x)=1_H\}=\{x\in V\,|\,T(x)=0_W\}=\ker T$$
(Here's a lot of notational changes going on; I hope it's not too confusing.)

I will just mention that the concept of a kernel can be generalized beyond this simple example to include the kernels of ring homomorphisms, or module homomorphisms and more related notions. This is best done using some category-theoretic language and universal properties to move to arbitarty categories. This general concept (a special case of a so-called equalizer) will be exactly the same if we then consider vector spaces and linear maps or groups and group homomorphisms. However, this requires some additional work as the definition is not in set-theoretic terms per se.
