MAT311-ASSIGNMENT 1 7 September 2018 QUESTION 1 Let Gand Hbe groups and let G H =f

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

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