(

Group Theory)

Given a

homomorphism `H` between two

groups

`G`_{1}
and

`G`_{2}, the homomorphism's "kernel" (symbolized

`Ker
H`) is the set of elements of

`G`_{1} that
give rise to the

identity element of

`G`_{2}.

If we call the identity element of `G`_{2} "`i`_{2}",
we can say

`g e Ker H -> H(g) = i`_{2}

`Ker H` necessarily contains the identity element
`i`_{1}
of `G`_{1}.

Since `H` is a homomorphism, for all `p, q e G`_{1},

`G`_{2}(H(p), H(q)) = H (G_{1}(p, q)).

Let `a = H (i`_{1}). Now,
for any `g e G`_{1},

`G`_{2}(H(i_{1}), H(g)) = H (G_{1}(i_{1},
g))

But this means that

`G`_{2}(a, H(g)) = H (g)

for all `g e G`_{1}, Although `Im H`
is not necessarily `G`_{2}, `G`_{2}
is still a group. Therefore, `a = i`_{2}, meaning
`i`_{1} e Ker H, which was to be proven.