Representation & Finite Groups

Representation & Finite Groups

REPRESENTATION THEORY OF FINITE GROUPS

Total marks of correct solutions is 30
marks.
1) i) Let G be a finite group and g ? G. Prove that if g is central
(that is xg = gx for all x ? G), then
|?(g)| = |?(1)|
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 x
2 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(x
2
) = 3, f(y) = f(xy) = 0,
g(1) = 4, g(x
2
) = 10, g(x) = g(y) = g(xy) = 0.
Is either of these functions the character of a representation? If yes,
then find the corresponding representation.
ii) Compute the ring R(D4) and the Adams operations therein.
3) Let G be the group of order 16 defined in terms of generators and
relations
G =< x, y : x
4 = 1 = y
4
, yx = x
3
y >
Find character table of G. Compute the ring R(G).
1
2 PROBLEM SHEET 2

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