MAT311-Tutorial 1
Q 1In the following, determine whether the systems described are groups. If they are not groups, point out which axioms fail to hold.
(a) (Z ; ), a b= a b
(b) (Q ; ), a b= ab
(c) (2 Z ; ), a b= a+ b
(d) (N ; ), a b= a+ b
Q2Check if the following are groups. If they are groups (include checking closure), show if they are abelian, non-abelian, nite or innite.
(a) nZ : multipliers of nby integers under addition.
(b) Z
n : integer modulo
nunder addition.
Q3(a)Make addition table for Z
5.
(b)Find the orders of elements of Z
5.
Q4Let ; ; 2S
6 such that
=
1 2 3 4 5 6
3 1 4 5 6 2
;
=
1 2 3 4 5 6
2 4 1 3 6 5
;
and =
1 2 3 4 5 6
5 2 4 3 1 6
:
Compute the indicated products.
; 2
; 2
;
1
MAT311-Tutorial 2
Question 1(a)Let Gbe a group and let a; b2G. Show that
i. (ab )2
= a2
b 2
if and only if Gis abelian.
ii. (ab )
1
= a
1
b
1
if and only if Gis abelian.
(b)In S
3, give an example of two elements
a; bsuch that (ab )2
6
= a2
b 2
.
(c)Is S
3 abelian? Give a reason to your answer.
(d)Find the orders of elements of S
3. Argue why
1
1 =
1.
Question 2(a)For D
3=
fe; r; r 2
; s; rs; r 2
s g , verify that (rs )r 2
= r2
s , (r 2
s )( rs ) = r
and sr2
= rs with r3
= e= s2
and sr=r2
s .
(b)Work out clearly (r 3
s )( r2
s ) and (r 2
s )( r3
s ), the elements of D
4.
(c)Find the order of r3
and r2
s in D
4.
Question 3(a)Find the cycle decomposition of the permutation =
1 2 3 4 5 6 7 8 9 10
3 6 5 7 8 9 10 1 2 4
;
(b)Express the permutation 2 S
10 as a product of transpositions.
(c)Find the order of
Question 4(a)Let Gbe a group and let a; x2G. Prove that (xax
1
)n
= xan
x
1
for n 1. Deduce further that xax
1
and ahave the same order.
(b)Show that the element
A=
1 1
0 1
2 GL (2; IR )
is of innite order.