# Answer to Question #45562 in Abstract Algebra for SAPNA

Question #45562

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.

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.

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