Advanced Abstract Algebra - Maharshi Dayanand University, Rohtak

UNIT-I
43
abab------ab = a n b n
−
bg n 1 n n
a ba b = a b
bg
n− 1 n− 1 n−
1
ba = a
b
To show G ( n−1)
∆ G.
i.e. To show aza
Let
(
z = x n −1
) , x ∈G
j
− 1
∈G n−
1
( ) , v a ∈ G, v z G
n
∈
( −1)
n−1
d i d i
−1 n−1 −1 −1
( n−1)
Now aza = ax a = axa ∈G Θ axa -1 ∈G
G (n-1) ∆ G.
(iii) To show a
n−1 b
n b
n a
n−1 = v a, b ∈G
(1)
Also
= − n − 1 −
i n
i n 1
n
n n
n
dd n
n
o{ d i }
it
di n i
n
i i
n
from (1) and (2)
b a(ba) 1 (ba) - - b - - - (ba)b −1
b= n
a
n
b− 1 −1 −1
( ba a
n
a -1
−)
1
n−
1
b n −( a −
= b = = b b −a 1 ban nb a b a n
−1
n
( aba
n (
−
n
1
− ) 1)
−1 −1
b
b a
n a
n −1
( )
= a b a
(2)
n−1 n−1
n
b n a = a b
(iv) To show
−1 −1
n n−1
daba b i ( ) = e v a, b ∈G
L.H.S. =
n−1
n
n n n
{ d i } e bg
−1 −1
n−1 −1 −1 −1
= ba b a Θ ba = a b from above
d
− −1 −1 n−1
n
( n )
= ba b a
=
i
j
=
=