Consider the nine groups listed below. Find 14 subgroup relations of the formGi ≤ Gj (i 6 = j) amongst them. [5 marks]G1 = M2,2(R) under matrix addition,G2 = all nonsingular 2 × 2 matrices with rational entries, undermatrix multiplication,G3 = all 2 × 2 matrices with integer entries, under matrix addition,G4 ={(0 00 0)}under matrix addition,G5 ={(1 00 1),(−1 00 −1)}, under matrix multiplication,G6 = all nonsingular 2 × 2 diagonal matrices with real entries,under matrix multiplication,G7 ={(1 00 1)}, under matrix multiplication,G8 = all 2 × 2 upper triangular matrices with integer entries,under matrix addition,G9 = all 2 × 2 matrices with real entries and determinant ± 1 undermatrix multiplication

Write do n all the subgroups of (𝑍6, +6) and (𝑍8, +8

C L R T B Q S M A P D I  N F J K G Y XFour of the following  five are alike in a certain way and so form group. Which is the one that does not belong to the group?1. LBT     2.IJF    3. PID    4. BMSQuestion 37Answera.3b.4c.1d.2

2. Consider the following matrices.[ 1 3−2 6](a)[1 −91 −5](b)1 2 0−1 −1 10 1 1(c)1 1 00 −2 10 0 3(d)1 1 01 0 10 1 1(e)1 1 −10 2 0−1 1 1(f)4 0 12 3 2−1 0 2(g)1 0 32 −2 23 0 1(h)3 1 0 0−1 1 0 00 0 1 40 0 1 1(i)2 1 1 10 1 2 30 0 3 30 0 0 2(j)1 0 0 00 1 0 01 1 3 0−2 1 2 −1(k)1 1 1 04 1 0 10 0 −1 10 0 2 0(l)For each of the matrix, find(i) the characteristic polynomial

Consider the direct product D8 × S4 where D8 is the group in Question 6 andS4 is the symmetric group (with its usual operation).(a) Find the product(µD,(1 2 3 42 4 3 1)) (ρ90,(1 2 3 41 4 2 3)).[2 marks](b) Find the inverse and the order of(ρ180,(1 2 3 42 4 3 1)).[4 marks]8. (a) State Lagrange's theorem and use it to show that if G is a group of orderp2, where p is prime, and H is a non-cyclic subgroup of G then G = H.[6 marks](b) Let H be a subgroup of a group G and let a, b ∈ G be such that aH ∩bH 6 =∅. Show that aH ⊆ bH. [4 marks]4

The dihedral group G = D8 has elements {ρ0, ρ90, ρ180, ρ270, µH , µV , µD, µU }.Its multiplication table is partially completed below.ρ0 ρ90 ρ180 ρ270 µH µV µD µUρ0 ρ0 ρ90 ρ180 ρ270 µH µV µD µUρ90 ρ90 ρ180 ρ270 ρ0 µU µV µHρ180 ρ180 ρ270 ρ0 ρ90 µV µH µU µDρ270 ρ270 ρ0 ρ90 ρ180 µU µD µH µVµH µH µU µV µD ρ0 ρ180µV µD µU ρ180 ρ0 ρ90µD µD µH µU µV ρ90 ρ270 ρ0 ρ180µU µU µV µD µH ρ270 ρ90 ρ180 ρ0(a) Calculate the missing entries. (You are only required to write down themissing entries, you do not need to copy out the table.) [3 marks](b) Give an example of a cyclic subgroup of G and an example of a non-cyclicsubgroup of G. [3 marks](c) Show that K = {ρ0, ρ180} is a subgroup of G. [2 marks](d) Find all the right cosets of K = {ρ0, ρ180} in G.