1.    (i)  Determine which of the following are groups under the stated operations. For those which are groups, state the identity and inverses, and for those which are not, give one axiom that fails.

(a) The set of 2 x 2 non-singular matrices, under addition,

(b) The set of positive real numbers, under division,

(c) {0．2．4．6．8{, under addition modulo 10,

(d) {2．4．6．8．10．12{ under multiplication modulo 14.

(ii)  Define a subgroup of a group 点, and prove that the intersection of any two sub- groups of 点 is a subgroup.  (You may assume without proof that a subgroup has the same identity as 点, and the inverse of an element of the subgroup is the same evaluated in 点 or the subgroup.) Give examples of subgroups 力 and K of (Z．+) whose union is not a subgroup, and calculate what 力 ~ K is.

2.  (i) Define the order of a group, and the order of an element of a group.

(ii) Prove Lagrange’s Theorem, that the order of a subgroup of a finite group 点 divides the order of 点.

(iii) The group table of the quaternion group ○ of order 8 is given.  Find the orders of all elements of ○, and find all the right cosets of the subgroups 力 = {1．_1{ and J = {1．_1．j．_j{.

1    _1       i    _i      j    _j      k   _k

_j

j

_k

k

1

_1

i

_i

3.    (i)  Define a permutation of a set x .   Let f and g  be the  permutations of x  = {0．1．2．3．4．5．6{ given by f (z) = z + 3 and g(z) = 3z, where each is evaluated modulo 7. Write f and g as products of disjoint cycles.  Also express f2 , g2 , fg , and gf as products of disjoint cycles, and for each of f , g , f2 , g2 , fg, and gf , determine whether it is even or odd.

(ii)  Prove that the  mapping  θ from a group  点 to  itself given  by θ(g)  =  g2   is a homomorphism if and only if 点 is abelian.

(iii) State the first isomorphism theorem for groups.

By calculating the kernel and image of the map given in part (ii), deduce that R* /{1．_1{  R+ , where R*  and R+  are the sets of non-zero and positive reals respectively (and the operation is multiplication).