Math 210A Homework 6
Question 1. Let G-Sets be the category of sets with a G-action (objects) and G-equivariant
maps (morphisms). Let T:Sets −→ G-Sets be the obvious “trivial G-action” functor.
(a) Prove that the functor (−)G:G-Sets −→ Sets where YG={y∈Y|gy =y, ∀g∈G}
(the fixed points) is a right adjoint to T.
(b) Prove that L:G-Sets −→ Sets defined by L(Y) = G\Y={Gy |y∈Y}=Y/ ∼where
y∼gy,∀g∈G(the set of orbits of Yunder the G-action) defines a left adjoint to T.
Question 2. For Ga group, let G0= [G, G]. Let G(0) =Gand G(n)= (G(n−1) )0for n≥1.
(a) Show that Gis solvable if and only if there is an nwith G(n)= 1.
(b) Show that Gis solvable if and only if it has a normal tower with abelian subquotients.
(c) Is it true that any subnormal tower of a solvable group is automatically normal?
Question 3. For any group G, we define a chain of subgroups by setting Z0(G) = 1, Z1(G) =
Z(G), and Zi+1(G) to be the subgroup of Gcontaining Zi(G) such that Zi+1(G)/Zi(G) =
Z(G/Zi(G)). Such a chain is called an upper central series. A group Gis said to be nilpotent
if Zc(G) = Gfor some c∈N.
(a) We call a normal tower 1 = H0CH1C···CHm=Gcentral if Hi+1 /Hi⊂Z(G/Hi)
for each i. Show that Gis nilpotent if and only if it admits a central normal tower.
(b) In Gnilpotent, is any normal tower necessarily central?
(c) Let 1 →N→G→H→1 be a central extension (i.e. N⊂Z(G)). Show that Gis
nilpotent if and only if Nand Hare, if and only if His.
(d) Can one remove “central” in statement (c)?
(e) Construct the lower central series of Gby setting G0=G,G1= [G, G], and Gi+1 =
[G, Gi]. Prove that Gis nilpotent if and only if there is a c∈Nsuch that Gc= 1.
(f) Show that any p-group is nilpotent.
(g) Show that nilpotent implies solvable, but that the converse is false.
(h) Show that the cartesian product of a finite number of nilpotent groups is nilpotent.
