1. (a) Determine which of the following are groups under the stated operations. For those

which are groups, show that the axioms hold and state the identity and inverses.

For those which are not, show that one of the axioms fails.

i. The pair ({2, 4, 6, 8}, 0) where xo y = xy mod 10.

ii. The set {a+b√3 a, b € Z} under addition.

iii. The set of all vectors in R³ and the operation is vector product.

(b) Show that the following are homomorphisms. Calculate their kernel and image,

showing your working. Giving reasons for your answers, which of the homomor-

phisms are injective and which are surjective?

i. 4: Z7 → Z7 given by y(x) = 3x mod 7.

{(89) | ª€ R₁, b€]

R*,be Rwith matrix multi-

plication and p(A) = det(A) for A € G. (Recall that R* = R\{0}.)

iii. : Z → S3 given by (n) = (123)" where S3 is the group of permutations

on 3 objects.

ii. p: G→ (R*, x) where G

=