Find the oentre of group 𝑆3 and quaternion group

Define oentre of the group and prove that it forms a subgroup

Which of the following groups are cyclic? Explain your answers.(a) µn, i.e. the complex n-th roots of unity under multiplication;(b) the group of symmetries of a rectangle;(c) the rational numbers Q under addition

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.

Show that the following relations for the group velocity are true: 𝑣𝑔 = −𝜆2 𝑑𝜈𝑑𝜆;

The Group 15 element which does not show allotropy is ____________________