5.10 Example: Group Algebras

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In this short section we give an example of an important class of rings for

which the conclusion of Theorem 5.9.5 holds. To this end, let G be a finite

group, and let F be a field. Define the F-group ring FG, by setting

FG = {functions _ : G ! F}.

The operations are pointwise addition and convolution multiplication. Thus,

if _, _ 2 FG, and if g 2 G, then

(i) (_ + _)(g) = _(g) + _(g),

(ii) (_ _ _)(g) =

X

h2G

_(gh−1)_(h).

Note that we may identify g 2 G with the characteristic function in FG

on the set {g}, viz.,

g(h) =

_                           

1 if h = g

0 if h 6= g.

Thus, we may write _ 2 FG as _ =

X

g2G

_(g)g, and the convolution multiplication

is simply the ordinary group multiplication, extended by linearity.

As a result, we can think of elements of FG as F-linear combinations of

elements of G.

The ring A := FG is actually an F-algebra in the sense that it is not only

a ring, but is an F-vector space whose scalar multiplication satisfies

_(ab) = (_a)b = a(_b),

_ 2 F, a, b 2 A. Thus we often call FG the F-group algebra.

Let G be a finite group, and letM be an F-vector space. A representation

of G on M is a homomorphism _ : G ! GLF(M).1 Note that this gives M

the structure of an FG-module via

X

g2G

_gg · m :=

X

g2G

_g_(g)m,

m 2 M. Conversely, if M is an FG-module, then we get a representation

of G on M in the obvious way. Note that if dim M = n, then choosing

1Of course, this makes perfectly good sense even if G is not finite.

5.10. EXAMPLE: GROUP ALGEBRAS 147

a basis of M induces a homomorphism G ! GLn(F). Conversely, such a

homomorphism clearly defines a representation of G on M.

The main result of the section is this:

Theorem 5.10.1 (Maschke’s Theorem) Let G be a finite group, and let

F be a field whose characteristic doesn’t divide |G|. Then any FG-module

is semisimple.

Thus we see that if char F/ |G|, then FG satisfies the conditions of

Theorem 51.

In case the field F satisfies the condition of the above theorem and is

algebraically closed we can make a very precise statement about the structure

of FG.

Theorem 5.10.2 Let G be a finite group and let F be an algebraically

closed field of characteristic not dividing |G|. Then there exist integers

n1, n2, . . . , nt with FG _= _t

i=1Mni(F).

We mention in passing that even in the non-semisimple situation, i.e.,

when the characteristic of F divides the order of the group G, then it still

turns out that finitely-generated modules over the group algebra FG are

projective if and only if they are injective. (See, e.g., C. W. Curtis and I.

Reiner, Representation Theory of Finite Groups and Associative Algebras,

Wiley Interscience, New York, 1962, Theorem (58.14).)

Exercises

1. Let C be the complex field, and let A be an abelian group. Prove that

any irreducible CA-module is one-dimensional.

2. Let A be a cyclic group of order 3, say A = hti, and let F = F2, the

field of 2 elements. Prove that the assignment

t !

_

0 1

1 1

_

2 M2(F)

defines an irreducible representation of G.

3. Let G be a finite group, let F be a field and let _ : G ! F× be a

homomorphism. Define e =

1

|G|

X

g2G

_(g−1)g 2 FG.

148 CHAPTER 5. MODULE THEORY

(i) Prove that e is an idempotent.

(ii) Prove that if A = FG, then dimFAe = 1. (Thus Ae is a minimal

left ideal of FG.)

4. Let G be a p-group, where p is a prime and let F be a field of characteristic

p. Interpret and prove the following: The only irreducible

FG-module is the trivial one. (Hint: Let G act on the vector space M

and let z 2 Z(G) have order p. Let M0 = {m 2 M| z(m) = m}, and

argue that 0 6= M0 _ M. Next, show that M0 is a sub-FG-module

and so if M is irreducible, M0 = M. Thus z acts trivially on M; this

makes M into a F(G/hzi)-module. Now apply induction.)

5. Let G be a finite group, and let F be a field of characteristic p, where

p||G|. Prove that A := FG is not semisimple. (Hint: consider the

element a =

P

g2G g, and show that a2 = 0. Next, argue that the left

ideal I = FGa is one- dimensional and is equal to {_a| _ 2 F}. If A

is semisimple, then A = I _ J, for some left ideal J _ A. Now write

1 2 A as 1 = _a + _, where _ 2 J. What’s the problem?)

Chapter 6 Ring Structure Theory

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