Mathematics 667
A. Hulpke
Homework (due Apr 11)
47) Let G be a finite group and g ∈ G.
a) Describe the irreducible characters of ⟨g⟩.
b) Suppose for every p up to the maximal element order in G the p-power map on the conjugacy
classes of G, that maps a class C to the class of its p-th powers is known. Give a formula that gives
the values for χ ↑G⟨g⟩ for all χ ∈ Irr(⟨g⟩). (Such a formula is implemented in GAP by the command
InducedCyclic(c,"all");)
c) Determine the characters described in b) for the alternating group A4 .
48) Let G be a finite group and N ⊲ G, χ ∈ Irr(G) and 1N the trivial character of N. Show:
If (1N , χ ↓N ) =/ 0, then N ≤ ker(χ).
49)
by
The head of the character table of SL2 (9) (i.e. element orders and centralizer orders) is given
2
3
5
4
2
1
1a
4
2
1
1
2
.
1
2
.
1
2
.
1
2
.
1
.
1
1
.
1
1
.
1
1
.
1
3
.
.
3
.
.
3
.
.
2a 6a 6b 3a 3b 5a 5b 10a 10b 8a 4a 8b
a) Suppose S4 ≤ SL2 (9). What are the (two) possibilities for fusion maps (that is, the map that for
each class of the subgroup assigns the class of the group in which it lies) from S4 to SL2 (9)?
SL (9)
b) For both fusion maps obtained, and χ, ψ the degree 3 characters of S4 , compute χ ↑S4 2 and
SL (9)
ψ ↑S 4 2 .
SL (9)
SL (9)
c) For both choices of the fusion map, compute (χ ↑S4 2 , ψ ↑S4 2
d) Conclude that SL2 (9) has no subgroup isomorphic to S4 .
).
52) Let G be a group and χ, ψ be permutation characters. Show that χ + ψ and χψ both are
permutation characters. What are the corresponding actions?
(This shows that the set of all Z-linear combinations of permutation characters forms a ring. It is
called the Burnside-Ring of G.)
53) Let χ1 , ⋯, χ k be characters of (not necessarily irreducible) representations and let ϕ = ∑ a i χ i
with a i ∈ Z a class function such that (ϕ, ϕ) = 1 and ϕ(1) > 0. Show that ϕ is an irreducible character.
Problems marked with ∗ are bonus problems for extra credit.