Note: This is a soft-question.
Last week, my PhD supervisor spent a good 45 minutes going over Steinberg endomorphisms and twisted groups of Lie type. There's a chance they'll be important for my research, should the direction of my project go down a likely path.
There was a moment when I noticed a similarity between the way these groups are constructed and the way semidirect products are. I mentioned it to him at the time but we were towards the end of the session. We didn't go back to it.
All I have to go on are pictures I took of the blackboard, my memory, a vague understanding, and what I can gather from the internet so far. Also, I am reading Carter's, "Simple Groups of Lie Type"; I finished the third chapter recently.
The analogy I'd like to make more precise goes something like this:
- The homomorphism that defines a semidirect product might "collapse" in such a way that you're left with a direct product, not just a semidirect product.
- The Steinberg endomorphism that defines a group of Lie type might "collapse" in such a way that you're left with a group that is not twisted.
To me, this seems less than satisfactory as a precise comparison. My thoughts are difficult to articulate here. This is where I need your help.
The Question:
Can we make more precise an analogy between semidirect products and twisted groups of Lie type?
Further Context:
Steinberg endomorphisms are new to me. It should be no surprise, then, that twisted groups are new to me too; before the session, I was aware of twisted groups and had a rough idea of how they worked though.
Finite groups of Lie type turn out to be ideal test subjects/candidates for most of my research ideas lately, according to my supervisor. The more I learn about them, the more I agree. Thus, it is likely that I will come back to answers to this question in the future.
Please pitch your answers at a beginning graduate level. Assume familiarity of semidirect products but not with groups of Lie type.
I prefer to think of semidirect products in terms of their presentations. My go-to examples are the dihedral groups.