## Math 503: Abstract Algebra

Homework 4 <me@tylerlogic.com> In Collaboration With
Keaton Naff
February 22, 2014
http://coursework.tylerlogic.com/courses/upenn/math503/homework04

### 1

Let G be a finite group and S be a finite set on which G acts, and the action is given by μ : G × S S. Let V be the space of valued functions on S. Let ρ : G GL(V ) be the linear representation of G defined by

Let χρ be the character of (V,ρ).

Let the elements of S be {s1,,sm}, and denote by = {fsi}⊂ V the basis of functions such that fsi(sj) = δij for each sj S.

#### (a) Prove that (χρ|1G) is the number of conjugacey class in G

Fix an s S, and define f to be the vector of V by

Then for any g G and sj S

However, if g-1sj Gs, then there exists a g′∈ G with g-1sj = gs, which implies sj = ggs, meaning that sj is also in Gs. Hence the above equation boils down to

Thus the 1-dimensional subspace Ws = span{f} is G-stable. Since there is one such Ws corresponding to each conjugacey class Gs, then (χρ|1G) is the number of conjugacey classes. 1

#### (b)

Now let G operate transitively on S, and define U V be the set of functions f for which sSf(s) = 0. Then for any f U, f = a1fs1 + + amfsm

implying that a1 + + am-1 = -am, which further implies that U has dimension dimV - 1.

Now because G is transitive, then for any g G

implying that ρg(f) U, i.e. U is G-stable. Therefore V can be decomposed into V = U W for some G-stable subspace W, by Maschke’s theorem. However, since U has dimension dimV - 1, then W has dimension 1, and since G is acts transitively on S, then (χρ|1G) = 1, implying that W must be the single trivial representation in the decomposition of V . Thus, denoting the character of U by χU,

### References

[Sho]   Clayton Shonkwiler. Algebra hw 2. http://www.math.uga.edu/~clayton/courses/503/503_3.pdf.