# $AGL(V) = V \rtimes GL(V)$ with $GL(V)$ acting from the right

For a vector space $V$, I have constructed $AGL(V) = V \rtimes GL(V)$ as the elements $(v, A) \in V \times GL(V)$ (Cartesian product of sets, not a direct product) with multiplication $(v, A) (w, B) = (v + Aw, AB)$. This then has a faithful action on $V$ given by $$x^{(v,A)} = A^{-1}(x + v)$$ which is fairly ugly.

It was suggested to me that letting $GL(V)$ act on $V$ from the right (that is, write $V$ as row vectors rather than column vectors) would be nicer, but I'm struggling to make a construction that is actually nicer.

I'm theoretically trying to get to an action like $$x^{(v,A)} = xA + v, \tag{\ast}$$ (this is the action that e.g. [DM96, section 2.8] and [Neu87, section 2] use, but neither spell out the full group) but every multiplication I define is either invalid with the group actions, seems to not be a valid multiplication for a semidirect product, or is ugly.

E.g.

1. Taking the obvious $(v,A)(w,B) = (v+wA, AB)$ does not work with $(\ast)$ (nor does it work with $(x + v)A^{\pm 1}$).
2. The "multiplication" $(v,A)(w, B) = (Bv + w, AB)$ does work with $(\ast)$, $$x^{(v,A)(w,B)} = (xA + v)^{(w,B)} = (xA + v)B + w = x(AB) + (vB + w) = x^{(vB + w, AB)}.$$ However, AFAIK, a semidirect product should be adjusting the first element of second term by (some image in $\mathrm{Aut}(V)$ of) the second element of the first time, that is $(a,b)(c,d) = (ac^b, bd)$, and this is the reverse: $(a,b)(c,d) = (a^d c, bd)$. (Maybe I am being too inflexible with the definition of a semidirect product?)
3. Defining $(v,A)(w,B) = (v + wA^{-1}, AB)$ works with $x^{(v, A)} = (x + v)A$, but this does not seem any clearer than my original one, and also doesn't match the natural action above.

I feel like I am missing something simple; could someone suggest a more natural representation & action of $AGL(V)$ (from either the left or right)? (I imagine there may be a canonical one that is so 'obvious' it is not worth mentioning in anything but the most basic text.)

[DM96] J. Dixon and B. Mortimer. Permutation Groups. Graduate Texts in Mathematics. Springer New York, 1996.
[Neu87] P. Neumann. “Some algorithms for computing with finite permutation groups”. In: Proceedings of Groups St Andrews 1985. Ed. by E. F. Robertson and C. M. Campbell. Cambridge Books Online. Cambridge University Press, 1987, pp. 59–92.

Using your first multiplication $\mathrm{AGL}(V)$ acts on $V$ from the left via:
$(v,A)*z:=Az+v$
This is an action since $$(w,B)*((v,A)*z)=(w,B)*(Az+v)=BAz+Bv+w=(Bv+w,BA)*z=((w,B)(v,A))*z$$ and feels fairly natural to me.
• Ah, of course, thanks. This is the opposite to my 2., so I have a mismatch between the right action $x^g$ and the natural way in which $\mathrm{AGL}(V)$ acts... this is annoying: I have a non-trivial essay using $x^g$, and that notation is the precedent in this field, so I don't think it feasible to switch to a left-action. Hm... – huon Oct 24 '14 at 10:48
• Well, the difference between a right and a left action is merely technical so it won't matter that much. On the other hand you can always think of the space $V$ as "rows" let $\mathrm{GL}(V)$ act on it from the right (in the "natural" way) and define your multiplication in $\mathrm{AGL}$ accordingly to get a nicer action from the right... – Sebastian Schoennenbeck Oct 24 '14 at 11:39