I had come across following Problem given by :
[I know here is same question But i am not convince with argument given By One of my Respected Teacher on MSE
Prove $G$ has normal subgroups of indexes 2 and 5
Suppose there is homomorphism from finite group G onto $Z_{10}$ Then G has Normal Subgroup of index 5 and 2.
I know By First Fundamental Theorem of Isomorphism $G/Ker \phi$ $\approx$ $Z_{10}$
So $10\space|\space|G|$.As $Z_{10}$ is abelian every subgroup is normal so $\left<2\right>$ and $\left<5\right>$ are normal subgroup .
$\phi^{-1}(\left<2\right>)$ and $\phi^{-1}(\left<5\right>)$ are normal in G. Order of $\phi^{-1}(\left<5\right>)$ is multiple of 2 i.e $2k$ and $\phi^{-1}(\left<2\right>)$ is $5m$
I had doubt that how this I can convert to show that they have required index normal group?
Any Help Will be appreciated