MA4142 REPRESENTATION THEORY OF FINITE
GROUPS
PROBLEM SHEET 2
This is the second problems sheet. The deadline for submitting so-
lutions is 3th December, 10.00. Total marks of correct solutions is 30
marks.
1) i) Let G be a nite group and g 2 G. Prove that if g is central
(that is xg = gx for all x 2 G), then
j (g)j = j (1)j
for every irreducible character .
ii) Show that if G has a irreducible representation : G ! GL(V )
such that Ker() = 1, then the center of G is a cyclic group.
2) The character table of D4 is given by
1 1 2 2 2
G 1 x2 x y xy
1 1 1 1 1 1
2 1 1 1 ??1 ??1
3 1 1 ??1 1 ??1
4 1 1 ??1 ??1 1
5 2 ??2 0 0 0
i) Let f and g be a class functions given by
f(1) = f(x) = 1; f(x2) = 3; f(y) = f(xy) = 0;
g(1) = 4; g(x2) = 10; g(x) = g(y) = g(xy) = 0:
Is either of these functions the character of a representation? If yes,
then nd the corresponding representation.
ii) Compute the ring R(D4) and the Adams operations therein.
3) Let G be the group of order 16 de ned in terms of generators and
relations
G =< x; y : x4 = 1 = y4; yx = x3y >
Find character table of G. Compute the ring R(G).
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2 PROBLEM SHEET 2