why are subgroups defined based on group homomorphisms rather than on its law of composition? i am getting reacquainted with Algebra after some time. I thought i had understood it the last time, but apparently not. The question vexing me is why are subgroups (like the kernel) of a group defined on the basis of a homomorphism rather than its internal law of composition? Is every subgroup of a group a kernel of some homomorphism? Is it possible that a subgroup which is a kernel of some homomorphism not a subgroup when it is evaluated on another homomorphism? Please forgive the naivette. Its difficult to grasp things when one is doing self-study.
Thanks,
Sachin

No, the definition of a subgroup is defined on the basis of its internal law of composition, and those subgroups which are defined on the basis of some homomorphism are defined as normal subgroups.
And there are a lot of subgroups which are not normal: for example, the group $A_5$ is simple; this means that there are no non-trivial normal subgroups of $A_5.$
Finally, yes, it is possible that the kernel of one homomorphism might not be the kernel of another homomorphism; e.g. $C_2\triangleleft C_4,$ where $C_n$ means the cyclic group of order $n,$ is the kernel of the homomorphism $f:C_4\rightarrow C_4$ defined as $f(x):=x^2.$ But $C_2$ is not the kernel of the homomorphism $g:C_4\rightarrow C_4$ defined as $g(x):=x^3.$
If you still have confusions, feel free to point it out!  
Hope this helps.
