Advanced Abstract Algebra - Maharshi Dayanand University, Rohtak

UNIT-I
13
Corollary 4: (Feremat's Little Theorem):
For every integer a and every Prime p, a p ≡ a (mod p).
Proof:
By division algorithm, a = pm + r, 0 ≤ r < p. Hence
a ≡ r (mod p). The result will be proved if we prove r p r (mod p). If r = 0, the result is trivial. Hence
which forms a group under multiplication module o p. Therefore by corollary 3,
r p-1 = 1. Thus r p r (mod p).
Normal Subgroups
If G is a group and H is a subgroup of G, it is not always true that aH = Ha,
Definition:
A subgroup H of a group G is called a normal subgroup of G if a H = Ha for every a in G. This is denoted
by H ∆ G.
Warning:
H
G does not indicate ah = ha
H ∆ G means that if
,
then ∃ some h 1
H such that
A subgroup H of G is normal in G if and only if x Hx −1
≤ H ∀ x ∈G.
∀a
∈r
hah
≡
∈ a = G{ ∈H
1h G , G. 1
2a
,,.
∀3 ,................. ∈Hd
Θ.
xHx − p
1
−⊆ 1}
H ∀x ∈G xH ⊆ Hx ∀x ∈Gi and
x − 1
Hx x − 1
H x − 1
−1
c⇔
b ∆Hagb = Hbgh Hc = bHab ⊆gH Hc hence = H Hx bab ⊆g Hc xH=
∀H x ∈b ab Ggc
d i , )
b g
cbgh cb gb gh
Factor = Ha groups bc = Ha(or Hquotient bc = Hagroups):
H b Hc ∀a, b, c ∈G.
Let H∆ g.The set of right (or left) cosets of H in G is itself a group. This group is called the factor group of
G by H (or the quotient group of G by H).
Theorem 2:
Let G be a group and H a normal subgroup of G. The set G H
= { Ha a ∈G} forms a group under the
operation (Ha) (Hb) = Hab.
Proof:
We claim that the operation is well defined. Let Ha = Ha 1
and Hb = Hb 1
.
Then a 1
= h 1
a and b 1
= h 2
b, h 1
, h 2
∈H.
Therefore, Ha 1
b 1
= Hh 1
ah 2
b = Ha h 2
b = aHh 2
b = aHb = Hab
(In proving this we used Ha = H a H and H G).
Further He = H is the identity and Ha -1 is the inverse of Ha, ∀ a
(Ha) (He) = Hae = Ha, and Ha Ha -1 = Ha a -1 = He = H,
G.