Advanced Abstract Algebra - Maharshi Dayanand University, Rohtak

42
ADVANCED ABSTRACT ALGEBRA
L.H.S.=
= τ -1 -1 = τ ∈
( ca) , d ) cb d a,
H
c -(ca) (c + + -ad+cb+d
(iii)
Example 8.
from (i) and (ii).
Let G be a group in which, for some integer n>1, (ab) n = a n b n for all a, b
∈ G
. Show that
i.
ii.
iii.
iv.
is a normal subgroup of G.
is a normal subgroup of G.
v a, b ∈G.
v a, b ∈G.
Solution:
(i) First we show G (n) is a subgroup of G.
Let
−1 n n −1 n −1 n −1
n
Now ab = x ( y ) = x ( y ) = ( xy ) , ∈
Θd y x g i -1
∴G
a a,
∴ g zN
τ
ad
(
c
is a subgroup of g.
To show G ( n)
∆ G.
i.e. To show a z a
−1
∈G ( n)
v a ∈ G, v
G ( n)
z ∈ z = x , x ∈G.
−1 n −1 −1
n
n
aza = ax a = ( axa ) n is an integer >1
Θb
.
g
(ii) To show G (n-1) is a subgroup of G.
(n-1) n-1 n-1
Let a, b ∈G , then a = x , b = y , x, y ∈G.
−1 n −1 n −1 −1 n −1 −1 n −1 −1 n −1 ( n −1)
ab = x ( y ) = x ( y ) = ( y x)
∈G x ∈G
b g
is a subgroup of G.
n n n
eΘ ab = a b v a, b ∈ G,
for some integer n>1.
Θd
y -1
i