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The a-s-s-e-s-s-m-e-n-t willbe 6 calculation questions.Please see the questions shown in the screenshot. I will send you all the info after being hired, eg PPTs, student access etc. Please send a draft in 12hrs -1 day time, day 2, and day 3 as well. + Will need to draft some questions to ask the teacher and revise base on feedback (Send bk ard in 1 day max)
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MATH2301 Assignment 1
(1) (6 marks) Use the extended Euclidean algorithm to determine every integer x in the range
−2000 ≤ x ≤ 2000 such that 258x ≡ 1 (mod 1105). Show your working.
(2) (5 marks) Find the smallest positive integer x such that
x ≡ 4 (mod 7),
x ≡ 3 (mod 8),
x ≡ 2 (mod 15).
Show your working.
(3) In this question you may write just a to denote the equivalence class of a modulo 45.
(i) (4 marks) List the elements of Z∗45 and give the order of each element. No explanation
required.
(ii) (3 marks) List the elements of a subgroup H of Z∗45 such that H has order 8. No
explanation required.
(iii) (2 marks) List the elements of Z∗45 /H (where H is as defined in Part (ii)). No explanation
required.
(4)
(i) (5 marks) Let n ≥ 3 be an integer, and consider the dihedral group Dn as defined in the
course notes. It can be shown that G = {1, σ, σ 2 , . . . , σ n−1 } is a subgroup of Dn and that
G is isomorphic to the cyclic group Zn . It can also be shown that if H is any subgroup
of G, then the subgroup of Dn generated by H ∪ {τ } has order 2|H|. Prove that for all
n ≥ 3, the dihedral group Dn has a subgroup of order d for each divisor d of |Dn |.
(ii) (4 marks) Show that each of the groups S2 , S3 and S4 has the property that if d divides
the order of the group, then the group has a subgroup of order d. You only need to specify
subgroups of the required orders, you do not need to prove they are subgroups.
(iii) Lagrange’s Theorem states that if H is a subgroup of a finite group G, then the order of
H divides the order of G. However, it is not true that every finite group G has a subgroup
of order d for each divisor d of the order of G. Whilst it is true (as we have seen) that
if G is a finite cyclic or dihedral group, or if G ∈ {S2 , S3 , S4 }, then G has a subgroup of
order d for each divisor d of the order of G, the same result is not true for the alternating
group A4 .
(a) (4 marks) Show that if H is a subgroup of index 2 in a group G, and g ∈ G has odd
order, then g ∈ H.
(b) (2 marks) Use the result from (a) to prove that A4 has no subgroup of order 6.
(5) (10 marks) Show that if n and p are positive integers such that p is prime and φ(n) = 2p,
then 2p + 1 is prime.
QUESTION 6 IS ON THE NEXT PAGE
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(6) Recall that the set of all invertible n by n matrices with entries from R forms a group under
(matrix) multiplication. This group is called the general linear group of degree n over R, and
is denoted by GLn (R). The notation GL(n, R) is also used for this group, and will be used in
this question.
Instead of having matrices with entries from R, we can instead have matrices with entries from
an arbitrary field F , and carry out all our calculations in F rather than R. That is, the entries
in the matrices are elements of a field F and matrix multiplication works in the same way,
except that matrix entries are added and multiplied in F rather than in R.
Many of the concepts, definitions and results for matrices with entries in R carry over in a
natural way to matrices over F . For example, we have det(AB) = det(A) det(B) – noting
that the determinant of a matrix over F is an element of F (so the product det(A) det(B) is
a product of two field elements). Just like when working over R, a matrix over F is invertible
if and only if its determinant is non-zero (that is, the determinant is not equal to the additive
identity 0 of F ).
The set of all invertible n by n matrices with entries from a field F forms a group under (matrix)
multiplication. This group is called the general linear group of degree n over F , and is denoted
by GL(n, F ).
(i) (3 marks) Let F be a finite field of order q. Determine the order of the group GL(2, F ).
(ii) (2 marks) Let n ≥ 2 be an integer, let F be a field, and let
H = {A ∈ GL(n, F ) : det(A) = 1}.
Show that H is a normal subgroup of GL(n, F ). The normal subgroup H of GL(n, F ) is
called the special linear group of degree n over F , and is denoted by SL(n, F ).
(iii) (2 marks) Let F be a finite field of order q. Determine the order of the group SL(2, F ).
(iv) (4 marks) Let SZ = {λI : λ ∈ F, λn = 1} where I ∈ SL(n, F ) is the identity matrix.
Show that SZ(n, F ) ⊴ SL(n, F ).
(v) (4 marks) We have seen that SZ(n, F ) ⊴ SL(n, F ). The quotient group SL(n, F )/SZ(n, F )
is the projective special linear group, and is denoted by PSL(n, F ). Determine the orders
of PSL(2, F7 ) and PSL(2, F8 ). Show your working.
( Interesting facts: For all n ≥ 2 and for any field F , the projective special linear group
P SL(n, F ) is a simple group, except that P SL(2, F2 ) ∼
= S3 and P SL(2, F3 ) ∼
= A4 are not
∼
∼
simple. The groups P SL(2, F4 ) = P SL(2, F5 ) = A5 and P SL(2, F7 ) are the two smallest
non-abelian simple groups. )
END OF ASSIGNMENT
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