MAT301 Assignment 2
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MAT301 Assignment 2
(1) Let G = O(2). Let l1 , l2 be two lines through the origin such that l2 makes angle α with l1 as in the picture. Let R8 be rotation by β . Let F1, F2 be reflections in lines l1 , l2 respectively.
Describe (F2R8F1R8(−)5 )33 F1 geometrically as either a rotation by a specific angle or a reflection in a specific line.
2
α
(2) Let A = -1(0) 0(1)、.
Find the centralizer of A in E(2). (Recall that E(n) denotes the group of all symmetries of Rn ).
(3) Does D10 contain a subgroup of order 4? Justify your answer.
(4) Let G = (z, +). Let a, b, c e z
(a) Prove that gcd(a, b, c) = gcd((gcd(a, b), c)
(b) Prove that〈a, b, c( =〈gcd(a, b, c)(
(5) The group of quaternions o consists of 8 elements 干1, 干i, 干j, 干k with the following multiplication table
Caley table for o
Find all proper subgroups of o and show that they are all cyclic.
(6) Let G be a finite group and let H, K be subgroups of G.
is it always true that HK = {hk ), h e H, k e K} is a subgroup of G? If yes, prove this, if no, give a counterexample.
(7) Find Z(SL(2, R)).
Hint: Let A e Z(SL(2, R)) and let B e SL(2, R) be a diagonal matrix. Look at the equation AB = BA.
(8) Let l1 be the line through 0 making angle α with the x-axis and let l2 be the line making angle β with the x-axis. Let F1, F2 be reflections in lines l1 , l2 respectively.
Using matrix multiplication verify that F1F2 = R2(a −8)
2022-10-08