Algebra

1. Let P be the category whose objects are ordered pairs of groups (A,B) and the morphisms in

Hom((A,B),(C,D)) are pairs of group homomorphisms (a,~) where a: A---+C , ~: B---+D .

a) Show that P is a category.

b) Let F assign objects in Grp (the category of groups) to objects in P by F(G) = (G,G) and

assign each morphism 8: A---+B to the morphism (8,8) in P. Show that F is a functor.

2. Let R be an integral domain. Since R is commutative, TornR (A, B) is an R-module. (You don't

have to prove this.) Prove : If A is torsion, then TornR (A, B) is torsion.

[Hint: Use induction on nand consider an exact sequence 0---+N---+P---+B---+0 where P is projective.

Also note that Torit(A,B);::: A ®R B]

3. Exercise 20 in 17.1 (p. 794)

4. Let G be a finite group with identity e, A an abelian group, and 8: G---+Aut(A) a homomorphism.

a) Show that (via 8) A is a ZG-module: (L>gg) • a= 2::>/~(g)(a) geG geG

For n:::: 1, let P n be the free Z-module with basis { (go,g1, ... gn) I gi E G} .

Pn is a ZG-module viaCl::>gg) • (g0 ,g1 , ••• ,gn) = L:ng (gg 0 ,gg1 , ••• ,ggn). (You don't need to show this.)

b) Show that p n is a free ZG-module with basis { (go,gl, ... gn) I gi E G} . n

c) For n:=::1 , define dn(go,gi, ... gn) = l::C-1)'(g0, ... ,gP ... ,gn), where" k" means remove gi.

dn d1 do

Show: ... ---+Pn---+Pn-l---+ ... ---+P 0 ---+Z is a free ZG-module resolution of Z , where d 0 (Lngg) = l:ng geG geG

From part c), we can construct Ext;c(Z,A) for any ZG-module A. In group theory, Ext~G(Z,A) is denoted

Hn(G,A). Recall that we started all this with a group homomorphism 8: G---+Aut(A), so we have Ax Ia G. Cor. 34 on p. 821 explains how H 1(G,A) reflects the structure of Ax Ia G.

d) Compute H1 (~,Z4) where O·a =a and 1·a =-a for all aE Z 4 and explain how Cor. 34

applies, noting that z4 X Ia z2 is isomorphic to the symmetries of the square.