Does associativity imply bijectivity in group operations?

[fresh_42]

By the way, when you say symmetry group, do you me the maximal (I think) order group for X?

I mean the group of all bijective functions $X \longrightarrow X$.

 

[valenumr]

I mean the group of all bijective functions $X \longrightarrow X$.

Ok, I think that's in line with what I thought, basically Sn from my material.

 

[valenumr]

Ok, I think that's in line with what I thought, basically Sn from my material.

Eh, let me walk that back a little. I mean, I'm still working on things where group operators are essentially scalar operations. I'm not sure what to expect yet when it goes beyond that.

 

[valenumr]

Are you/we distinguishing between a "group operator" and a "group action"?

I thought about this for a while, because, as I've mentioned, pretty much all of the material examples I've encountered so far have been based on integers, with a sprinkling of reals, and mostly finite groups. Think like an advanced ninth grader or typical eleventh grader could probably easily handle the material.

In any case, my consideration would be a nice Persian rug I happen to own. It does have multiple symmetries, say left / right reflection, front back reflection, and top / bottom reflection.

So when I say "group operator", I'm thinking of the concept of one element of the group composed with another (single) element of the group. When you say "group action", which I expect is more appropriate, it's like rotating the carpet some multiple of pi / 2 on the floor, or maybe flipping it over. It affects the whole rug, but it is just applying the operator to every element.

I'm still trying to think about non-unitary stuff as well that might shrink my rug, for example. But I don't what to get ahead of the material.

In any case, my other comment about working with scalars is mostly a reflection. I suspect that whole "abstract" part of abstract algebra doesn't really care about the group elements or group operator, so long as the rules apply.