Advanced Abstract Algebra - Maharshi Dayanand University, Rohtak

UNIT-I
Q.20.
Q.21.
Q.22.
Q.23.
Q.24.
Show that Normality is not a transitive relation in a group G i. e. H ∆ K ∆ G / H ∆ G .
Show that S n
is generated by (12) and (1, 2, 3, ----------, n).
Find the product of
(1) (12) (123) (12) (23)
(2) (125) (45) (1, 6, 7, 8, 9) (15)
Which of the following are even or, odd permutations:
(1) (123) (13),
(2) (12345) (145) (15)
(3) (12) (13) (15) (25).
Prove that the cyclic group Z 4
and the Klein four-group are not isomorphic.
b
g
45
Q.25.
Show that the group
is isomorphic to the group if all
matric
over R of the form
Example 10.
Let H be a subgroup of G and N a normal subgroup of G. Show that H ∩ N is a normal subgroup of H.
Solution:
Let x be any element of and h be any element of H.
l
2Z
L
a2 f×
∩: × 2R bZ
N→ 2 R f (
To ) show
hxh h
x ax + b,
a ≠ 0q
− 1
HN ∈ H . ∩∈ 0 1
, H a N g∩
∆
0. H. NG x.
∈gHg H and xH
∈ N
bgn ∆ G
−1
= ⇔ N∈ bg H = G.
= s
NM O Q P ≠
-1 -1
x ∈H, h ∈H hxh ∈H,
N ∆ G, h ∈H ⊆ G hxh ∈ N
∴ hxh -1 ∈H ∩ N v
v
Example 11.
Let H be a subgroup of a group G, let
Prove that
(i) N(H) is a subgroup of G
(ii) H∆ N Hbg
(iii) N(H) is the largest subgroups of G in which H is normal.
(iv)
Solution:
bg
(i) Let g1, g 2 ∈N H .