1. (i)  Give the definitions of group and abelian group.

Prove that for any integer n ≥ 2, the set Zn(*)  of members of {1, 2, 3, . . . , n − 1} which are coprime with n forms a group  under the operation of multiplication modulo n.  (You may assume without proof that if m and n are coprime, then there are integers x and y such that mx + ny = 1.)

(ii)  Find which of the following are subgroups of Z3(*)0 , giving reasons:

(a) {1, 7, 11, 13},    (b) {1, 7, 13, 19},    (c) {1, 11}.

(iii)  Prove that if x, y, and z are elements of a group G such that xy = xz, then y = z .

(iv)  Give an example of a group G and elements x, y, and z of G such that xy = zx but y z .

2. (i) State Lagrange’s Theorem, and from it deduce that any group of prime order is cyclic.

(ii)  Define the direct product of two groups G and H .

(iii)  Prove that any group of order 4 is isomorphic to Z4  or Z2  × Z2 , but that these two groups are not isomorphic.

(iv)  List the right cosets of the subgroup {I, D} of the dihedral group D4  of order 8, whose group table is given.

I     R    R2      R3      H     V     D    D\

R3

R3

I

R

V

H

D\

D

3. (i)  Define permutation of a set X, and transposition, odd, and even.

Prove that the family of all even permutations of a set X forms a normal subgroup of the group of all permutations of X .

(ii) The following permutations f , g in the symmetric group S10 , are given in 2-row

notation:

f = 、,   g = 、.

Write each of f and g as a product of disjoint cycles, and state with reasons which of f , g , f-1 , g-1 , fg , gf are conjugate.

(iii)  Prove that N = {id, (1 2 3), (1 3 2)} is a normal subgroup of S3 , and that H = {id, (1 3)} is a subgroup of S3  which is not normal.

Give a specific homomorphism from S3  to R* , the group of non-zero real numbers under multiplication, of which N is the kernel. (You do not need to prove that the mapping you define is a homomorphism.)