Laplace Transform and Groups

[Gus123456789]

In my abstract algebra class, we are doing research into groups. Normally this would go under the algebra section, but the set I picked involves differential equations more. A group is any set paired with a binary operation (2 input elements give one output element) that is closed (all outputs are on the input list), associative (a~b)~c=a~(b~c), has an identity (id~a=a~id=a), and every element has an inverse (an element paired with its inverse gives the identity as an output). This must be true for all elements. The set I picked involves the "Laplace Transform." The Laplace transform does this:
[tex]L[f(x)]=\int_{x=0}^{\infty } e^{-sx}f(x)dx[/tex]
The set I use includes all functions f(s) for which [tex]L^{-1}f(s)[/tex] exists, and my operation will be
[tex]L[L^{-1}(f)L^{-1}(g)][/tex]
where f and g are functions operable upon by the inverse transform. This set is not a group, as many elements do not have inverse, but an appreciable number of subsets are indeed groups. I will update this thread every few days, as I work through the functions.
off topic- this is a great forum for math! It is too bad it seems so inactive... I will recommend it to my professors and classmates.

[Gus123456789]

After some review, I decided to limit my input to functions of the form [tex]f(s)=\frac{a}{b+cs}[/tex] where a,b,c are real constants, and a,c are not zero.
This is mostly because the rest were not closed. For example, take [tex]f(s)=\frac{1}{s^{2}}[/tex]
Then [tex]f(f(s))=L(x^{2})[/tex] which does not exist.
At this point, it would be wise to point out the operation symbols that I will use. [tex]L, L^{-1}[/tex] represent the laplace transform and inverse transform, and [tex]\alpha[/tex] represents my group function.

After plowing through it on paper, I have discovered that [tex](\frac{a}{b+cs}) \alpha (\frac{d}{e+fs})=\frac{-ad}{scf+bf+ce}[/tex] This is an important first step, finding the identity and inverse become easy after. At a later date, I will prove this with algebra.

[amit28it]

how L{delta(t)}=1
what is t-shift theorem in laplace n wt is its proof?
what is linearity property in laplace transform?