This is an open-ended question, as is probably obvious from the title. I understand that it may not be appreciated and I will try not to ask too many such questions. But this one has been bothering me for quite some time and I'm not entirely certain that it's completely worthless, which is why I've decided to ask it here.
Why are groups so immensely more important in mathematics and its applications than semigroups are? I know little of group theory, mathematics and its applications, so I cannot understand how much more important they are. But I know they are because I see how many more people are interested in groups than in semigroups.
I wouldn't ask this question if this fact didn't seem a little strange to me. I know that groups are associated with symmetries, or with automorphisms of structures. Cayley's Theorem tells me that every group can be seen as a set (closed with respect to taking compositions and inverses) of automorphisms of a structure with no constants, functions or relations, i.e. a plain set. I know that the automorphisms of a vector space, a module or a group form a group. Obviously, automorphisms are important.
Then we have inverse semigroups. These are less popular but this I can understand. They are associated with partial symmetries, by the Wagner-Preston Theorem. Partial functions do seem much less used than functions.
But then come semigroups. Just like in the two previous cases, there is an "embedding theorem", which says that every semigroup can be embedded in a semigroup of maps from a certain set into itself. And, analogously to the case of groups, the endomorphisms of a vector space, a module or a group form a semigroup.
This seems to say that semigroups are to endomorphisms what groups are to automorphisms. The "conclusion" would be that
$\frac{\mbox{the importance of semigroups}}{\mbox{the importance of endomorphisms}}=\frac{\mbox{the importance of groups}}{\mbox{the importance of automorphisms}}$
Assuming that endomorphisms are about as important as automorphisms, we get that semigroups are about as important as groups. My feeling is that the assumption is correct.
I understand that I must be oversimplifying something at some point. But what am I oversimplifying and where?
I realize that this is possibly a very dumb question, but my confusion is genuine.
Edit: I have realized, after reading your answers and comments, that I made the mistake of using a very vague term without even attempting to define it. The problem seems to be what importance is. Is it popularity or usefulness, or being used a lot, or interesting or being interesting, or something else, or a mixture of many traits? I'm not going to force my understanding of the word now. But if someone still wants to answer this question, perhaps it would be good idea if the answer contained an attempt at clearing this up. (Or maybe not. I'm not sure!)
Edit: A question similar to mine is dealt with here.