MAT311-ASSIGNMENT 1

7 September 2018

QUESTION 1

Let

Gand Hbe groups and let G H =f(g; h ) :g2 G; h 2Hg. Dene a

multiplication on G H by (a; b )(c; d ) = ( ac; bd )for all a; c2G and b; d2H.

Further let G

1=

f(a; e

H) :

a2 Gg.

(a)Prove that G H is a group under this multiplication. 5

(b)Prove that if Gand Hare both Abelian then G H is Abelian. 3

(c)Prove that G

1is a subgroup of

G H. 4

(d)Given that every group of order 6 is isomorphic to the cyclic group C

6

or the symmetric group S

3, determine

C

2

C

3.

3

QUESTION 2 Let

Gbe a group and let Hand Kbe subgroups such that none of the sub-

group is contained in the other.

(a)Prove that HK is never a subgroup in G. 5

(b)Prove that a group cannot be written as the union of two proper sub- groups. 5

QUESTION 3 Prove that

H=

1 a

0 1

ja 2 IR

:

is an Abelian subgroup of GL(2; IR ). 10

QUESTION 4 Consider the quotient group

Q=Z whose elements are of the form m n

+

Z

where mand nare integers. The representatives of Q=Z are rational num-

bers in the interval 0;1) . Determine the order of

(a) Q=Z . 2

(b)each m n

+

Z. 3

QUESTION 5

(a)Let

Gbe an Abelian group and let Hbe a subset of Gdened as fol-

lows, H=fa 2 G :o(a ) = n; n 2Zg. Prove that HG. 5

(b)Prove that the additive group of rational numbers is not nitely gen- erated. 5

QUESTION 6 (a)Determine whether the binary operation

gives a group structure on

the given set. If no group results, give the axiom(s) that do not hold.

(i) (2Z; ) given by ab= a+ b

(ii) (C ; ) given by ab= jab j 3

(b)Prove that a group Gis Abelian if and only if (ab )

1

= a

1

b

1

for all

a; b 2G. 5

(c)Let Gbe a group and suppose abc=efor a; b; c 2G. Show that bca=e.

2

(d)Prove that if Gis a nite group of even order, then there is an element

a 6

= 0 such that a2

= e. 3

(e)Let Gbe a group and suppose a2 G of odd order. Prove that aand

a

1

have the same order. Prove further that there is an element b2 G

such that b2

= a. 5

(f)In S

3, give an example of two elements

aand bsuch that (ab )2

6

= a2

b 2

.

What does this example say about S

3?

3