Answer to Question #45562 in Abstract Algebra for SAPNA

Question

Let D12 = h

x; y x
2
= e; y
6
= e; xy = y
1
x i.
a) Which of the following subsets are subgroups of D12? Justify your answer.
i) fx; y ; xy ; y
2
; y
3
; eg ii) fxy ; xy
2
; y
2
; eg iii) fx; y
3
; xy
3
; eg (4)
b) Find the order of y
2
. Is the subgroup hy
2
i normal? Justify your answer.
c) Let
D2n = h

x; yj x
2
= e; y
n
= e; xy = y
1
x i
Prove the relation
x
i
y
j
x
k
y
l
=
(
x
i
y
j+l
if k is even
x
i+k
y
l j
if k is odd.
Further, find all the elements of order 2 in D12.
d) Find two different Sylow 2-subgroups of D12.