Show that for an associative and commutative binary operation * the products a * a * b * a * b and b * a * a * a * b are equal.

Prove: if in a monoid every element x different from the identity satisfies x² = e, then the monoid is commutative.

Let S be a semi-group. We can extend S with an element e, which is not in S, to a monoid (S {e},*,e). How do we have to define the multiplication on S {e} to make this indeed a monoid with unit e? What happens if S contains already a unit?

Which of the two monoids on 2 elements, Z/2Z with addition or with multiplication, is the extension of a semi-group with a unit element?

Describe an algorithm that takes as input an n × n multiplication table and that checks for associativity and commutativity of the multiplication.

Show that the direct product of two monoids is again a monoid.

Find two submonoids of Z/6Z such that their union is not a submonoid.

If S_i is a submonoid of the monoid M_i, for i=1, 2, then S₁ × S₂ is a submonoid of M₁ × M₂. Prove this.

Suppose a₁/b₁, a₂/b₂, ... , a_n/b_n are elements of Q, and p is an integer greater than |b₁b₂ · · · b_n|. Show that 1/p is not contained in the submonoid of (Q,+,0) generated by a₁/b₁, a₂/b₂, ... , a_n/b_n. Prove that Q is not finitely generated.

Let X be a nonempty set. If M is a monoid with unit element e, then we can define a monoid structure on the set F of all maps from X to M as follows.

• If f and g are in F, then their product f * g is defined by
  (f * g)(x) = f(x) * g(x).

• The constant map x-> e is the unit element.

Prove this.

Let M be a cyclic monoid generated by the element c. Suppose that c² e, c² c⁶, and c⁴ = c⁸. With which C_k,l is M isomorphic?

Let M be the cyclic monoid generated by c and isomorphic to C_k,n. Write an algorithm that rewrites every power of c to one of cⁱ, where 0 i < k + n.

Suppose that f : M -> M' is a morphism. Prove that the image f(M) is a submonoid of M' and that the kernel of f, i.e., {m M | f(m) = e'}, where e' is the identity of M', is a submonoid of M.

Determine all monoids on three elements.

Prove that in the monoid (Z/nZ,·,1) an element m has an inverse if and only if gcd(n,m)=1.

Let M₁, M₂ be monoids. Prove that the invertible elements of M₁ x M₂ are of the form (m₁,m₂) with m₁ invertible in M₁ and m₂ invertible in M₂.

What are the invertible elements of C_k,n?

Is the following true? If G is a group of order n, and m is a positive divisor of n satisfying 1 < m < n, then G contains an element of order m.

Let G be a finite group. Show that each element of G appears exactly once in each column and each row of the multiplication table (also called Cayley-table) of G.

Let I be the identity matrix of size n, i.e., the n by n matrix with ones on the diagonal and zeros outside the diagonal. For any matrix A we denote by A^T the transposed matrix of A. Prove that the set O_n(R) = {A GL_n(R) | A · A^T = I} is a subgroup of GL_n(R).

Determine the left and right cosets of S₃ in S₄.

Determine the order of the element (1,2)(3,4,5) in S₅. Prove that, in general, the order of a permutation equals the lcm of the cycle structure of a disjoint cycle decomposition.

Let G be a group and H a nonempty finite subset of G closed under multiplication. Prove the following.

1. For h H, the elements h¹, h², h³, ... are not all distinct.
2. The identity element is in H.
3. Every element of H has finite order.
4. H is a subgroup of G.

Let G be a finite group of order m Let g G. Suppose that for each prime divisor p of m we have that g^m/p is not the identity. Prove that the group G is cyclic and generated by g.

Prove that the groups Z/2Z × Z/3Z and Z/6Z are isomorphic. Show that these two groups are not isomorphic to S₃.

Let G be a cyclic group with generator g.

1. Show that the map f : Z -> G, defined by f(z) = g^z is a morphism of groups.
2. Suppose that G has order n. Show that the map f : Z/nZ -> G, with f(z)=g^z is well defined and is an isomorphism of the (additive) groups.

On R we define the operation * by x * y = x + y - xy.

1. Is * commutative?
2. Is * associative?
3. Is there an identity element in R with respect to *?

Consider the additive group Z × Z.

1. Prove that this group is not cyclic, but can be generated by the elements (3,4) and (2,3).
2. prove that (a,b) and (c,d) generate the group if and only if ad - bc = 1 or -1.

Consider the monoid M_n(R), where the multiplication is the ordinary matrix multiplication. Which of the following sets are submonoids:

1. The set consisting of only the zero matrix.
2. The set consisting of only the identity matrix.
3. The set of all matrices with determinant 1.
4. The set of matrices with trace 0.
5. The set of upper triangular matrices.

Determine for every m {3,4,5} the integers k and l such that the submonoid of (Z/mZ,·,1) generated by 2 is isomorphic to C_k,l.

Prove that the monoid (Z/8Z,·, 1) cannot be generated by less than 3 elements. Prove that it can be generated by 3 elements.

Determine the invertible elements of the following monoids.

1. Z/2Z × Z/3Z.
2. The multiplication monoid of Q[X]/(X²).
3. The multiplication monoid of Z/16Z .

Let (M,*,e) be a monoid. Define a new multiplication *' on M by x *'y = y * x. Prove that (M,*',e) is also a monoid.

Consider the group U₂₆ of invertible elements in the monoid (Z/26Z, · , 1).

1. How many elements does U₂₆ have?
2. U₂₆ is cyclic. Find all its the generators.

1. Show that the map f : Z × Z -> Z, f(x,y) = x - 2y is a morphism of the additive groups. What is the image of this morphism?
2. Let G be a group and g G. Prove that the map f : Z -> G, f(k) = g^2k is a morphism of groups. What is the image of f if the order of g equals 6 or 7, respectively?
3. Determine all morphisms of the additive group Z/4Z to itself. Which of these are isomorphisms?
4. If f : G -> G' and h : G' -> G'' are morphisms of groups, then the composition h _° f : G -> G'' is also a morphism of groups. Prove this. Deduce furthermore that if G is isomorphic with G' and G' with G'', then G is isomorphic with G''.

Let G be a group of order 4. Prove the following statements.

1. If G contains an element of order 4, then G is cyclic and isomorphic to Z/4Z.
2. If G contains no element of order 4, then G is commutative and isomorphic to Z/2Z × Z/2Z.

Let p be a prime. Then the multiplicative group (Z/pZ)^* is cyclic. (This will be proved in Chapter 7.) Write an algorithm that determines a generator for Z/pZ^*. Determine all odd primes p < 10.000 such that 2 is a generator for (Z/pZ)^*. (It is a conjecture of Artin that there are infinitely many primes p for which 2 generates the group (Z/pZ)^*. However, up to January, 1998, it is not known if this is really true.)