Advanced Abstract Algebra - Maharshi Dayanand University, Rohtak

UNIT-I
b
= φ ( φ φ ) ( )
g h t
x
g
31
(associative)
d
n
φ e
is the identiy and φ
g
i
= ( φ g ) – 1
g − 1
φ g φ e = φ ge = φ g ∀ g ∈ G, and
φ
g
φ − 1 = φ φ
g g g – 1
=
e, hence ( φ ) –
= φ − 1 )
Thus g = nφ g : g ∈Gs is a group of permutations.
Define ψ :
g →φ
∀g∈G
i.e. ψ
g
If g = h, then
ψ is one-to-one:
If
s
then φ
∴
Θ
G
( n = A ψ
φh
g
→( = )
φ ( )
h φ ) =
g n
≥≅ h, G
5. ) n= h
g.
= ≥ φ
h
gh , 5
φ . gt ( Gx φψ ) g ∀(g) ( φxh = ∈ φψ tG)
(h) g = h, i.e. ψ is one-one.
ggh : = ∈φ g
φ,
h∴=
ψ is onto.
ψ is a homomorphism:
ψ
Hence
g
g
1
is trivial, so ψ is a function.
( e) = φ ( e)
or g e = he i.e.
h
by definition of ψ,
ψ
is an isomorphism and so
ψ
g
Remark:
g is called left regular representation of g.
Simplicity of A n
for
Definition:
A group is simple if its only normal subgroups are the identity subgroup and the group itself.
The first non abelian simple groups to be discovered were the alternating groups The simplicity of
A 5
was known to Galois and is crucial in showing that the general equation of degree 5 is not solvable by
radicals.
Theorem 20.
The alternating group A n
is simple if n ≥ 5.