Proving Kernel of Homomorphism is a Subset of G

[matrixer]

Im trying to understand this theorem of Proving Kernel of Homomorphism is a Subset of G My question is since the definition of the kernel itself is ker(f)={xEG;f(x)=e'} Here it is given as xEG => kernel should be a subgroup of G.so is the proof required for this?

 

[pka]

I did not try to quote the question. It is so confused the way it has been written.
The statement is incorrect. The so called 'proof' is confused beyond belief.

Here is the correct statement:
If each of \(\displaystyle \mathfrak{G}~\&~\mathfrak{H}\) and \(\displaystyle \phi:\mathfrak{G}\to\mathfrak{H}\) is a homomomrphism then \(\displaystyle \mathfrak{ker}(\phi)\) is a subgroup of \(\displaystyle \mathfrak{G}\).

The proof is simple, three lines at most. Show that if \(\displaystyle \{a,b\}\subset\mathfrak{ker}(\phi)\) then \(\displaystyle a\,b^{-1}\in\mathfrak{ker}(\phi)\).

 

[matrixer]

Thanks for the reply.The question was wrong,it is subgroup not subset since kernel is already a subset.
Is the current proof lengthy ?