T1: Lectures 15-20 Flashcards
Define the normal subgroup H of G
The subgroup H which is invariant under conjugation with members of the group G.
Given a group G, subgroup H and rep of G: (ρ,V), define the restriction of ρ to G
ρ|_H (h) = ρ(h)
Given a normal subgroup N of G, what other group can we form? What special rule does it follow?
The quotient group G ̃=G/N
g_1 N ⋅ g_2 N = g_1 g_2 N
Given a quotient group G ̃=G/N and a rep (ρ ̃,V), define the inflation of ρ ̃ from G ̃ to G
ρ ̃(g) = ρ(gN)
(ρ ̃,V) is then a representation of G
Give an alternate condition for the subgroup N to define an inflation
N is a subgroup of the kernel of ρ
Given a rep (ρ,V) of G, when is ρ said to be ‘faithful’?
If the kernel of ρ is trivial: ker(ρ) = {e}
How do the characters of a group and its inflation relate?
χ_ρ ̃ (g)= χ_ρ (gN)
How do the inner product of characters of a group and its inflation relate?
⟨χ_ρ,χ_ρ ⟩=⟨(χ_ρ ) ̃,(χ_ρ ) ̃ ⟩
Give an equivalent definition of the kernel of ρ using characters
Ker(ρ) = {g in G: χ(g)=d}
For dimension of V, d.
Given H is a subgroup of G, define the left coset of H in G.
gH = {gh : h ∈ H}
How do cosets and quotient groups relate? Given group G and subgroup H
G/H is the set of left cosets of G; this forms the quotient group if H is normal.
Let G be a finite group with rep W and let H be a subgroup with rep V. What are the two conditions for an W to be induced from V?
i. There is an H-subrepresentation V_0 of W that is isomorphic to V
ii. W = g_1V_0 ⊕ . . . ⊕ g_rV_0
for the set of g reps of left cosets.
How do the dimension of the subgroup and induced rep relate?
dim(Ind V) = [G:H] dim(V)
How do representations of a group induced from the same subgroup relate?
They are isomorphic
Define Frobenius reciprocity
Given a rep W of G induced from the rep V of H and any rep U of of G, we have an isomorphism
Hom_G (W,U) iso Hom_h (V,U)
