Groups

In this question, the set S is {2, 4, 6, 8}, and ×₁₀ is the operation of multiplication modulo 10.

The group G consists of the set S, under the operation ×₁₀.

Complete the Cayley table for G given in the Printed Answer Booklet.

State the identity element of G.

State the order of each non-identity element of G.

State, with justification, the order of H.

State also the identity element of H.

State all the elements of H of order 2.

Using the result of part (a)(iii), or otherwise, explain why all the (non-identity) elements

which do not have order 2 must have order 4.

State whether or not H is cyclic. You must give a reason for your answer.

The group G consists of the set S = {1, 9,17, 25} under ×₃₂, the operation of multiplication modulo 32.

Complete the Cayley table for G given in the Printed Answer Booklet.

Up to isomorphisms, there are only two groups of order 4.

• C₄, the cyclic group of order 4

• K₄, the non-cyclic (Klein) group of order 4

State, with justification, to which of these two groups G is isomorphic.

List the odd quadratic residues modulo 32.

Given that n is an odd integer, prove that n⁶ + 3n⁴ + 7n² ≡ 11(mod 32).

For each element x in C show that it has an inverse element in C.

Explain why all groups of even order must contain at least one self-inverse element (that is,

an element of order 2).

Prove that any group, in which every (non-identity) element is self-inverse, is abelian.

A group G has order 4 n + 2, for some positive integer n, and i is the identity element of G. Let x and y be two distinct, non-identity, self-inverse elements of G. By considering the set

H = { i, x, y, xy }, prove by contradiction that not all elements of G are self-inverse.