Note: All assignments in Hungerford, unless otherwise noted. Also, assignments are to be turned in and will be graded, unless otherwise noted.

1. Show that every semigroup G can be embedded in a monoid. Hint: Let G^~ be the disjoint union of G and the one-point set {e} and define a suitable multiplication on G^~.
2. Show that every monoid is isomorphic to a submonoid of the monoid of self-maps of some set. Hint: Let the monoid "act on itself".
3. Deduce from (1) and (2) that every semigroup is isomorphic to a subsemigroup of the self-maps of some set.

1. Show that the result of the last problem (section 3, problem 9) is false if you drop the assumption that the group is commutative. In other words, give an example of a group G with two elements of finite order whose product is NOT of finite order. (Hint: let the group G be the group of self-maps of the real line R generated by translation (x --> x+1) and by reflection (x --> -x).)

1. Show by example that in the situation of Exercise 13 in section 4, there can indeed be more than one subgroup of order q. (Hint: there is already an example with q = 2, p = 3.)

1. What is the coproduct in the category of abelian groups? Justify your answer.

1. Strengthen the conclusion of problem 10 in II, 1 by showing that every finitely generated subgroup of Q is cyclic. (Prove this first; it's the key to 10a.)
2. Let x = (x₁, ..., x_n) be an element of the free abelian group Zⁿ of rank n. Show that there is a basis of Zⁿ with x as its first element if and only if x is not a multiple ky of any other element y of Zⁿ, with k > 1, or if and only if the GCD of the integers x₁, ..., x_n is 1.

1. Show that every group of order 54 has a non-trivial semidirect product decomposition. Deduce that every group of order 54 is solvable. Is every group of order 54 nilpotent? If so, is every group of order 54 abelian? Generalize to groups of order 2p^r, p an odd prime, r a positive integer.
2. Suppose that G is a group with a normal subgroup N such that N has trivial center and every automorphism of N is inner. Deduce that G splits as a direct product N x H for some subgroup H (isomorphic to the quotient group G/N). Hint: to construct H, you need to show there is a canonical way to lift elements of the quotient group G/N back up to G. So first show that each element of the quotient group lifts to a unique element of G that commutes with N. To do this, first lift the element to some element of G, then adjust this element by something in N.
3. Show that every finite group G of order > 2 has a non-trivial automorphism. Hint: if G is non-abelian, it has a non-trivial inner automorphism. Consider the abelian case separately, using the structure theory of finite abelian groups.
4. Show that any group of order 1225 = 49^.25 is abelian, and then classify such groups up to isomorphism. Hint: show that the Sylow subgroups are normal.
5. Show that (Z_p)ⁿ, an elementary abelian p-group, has no non-trivial characteristic subgroups. (Here p is a prime.)

1. Mimic the proof that the Gaussian integers are a Euclidean domain to show that the subring R of the complex numbers generated by Z and by a primitive cube root w = (-1 + sqrt(3) i)/2 of unity is a Euclidean domain. (Hint: The "absolute value" function is again the square of the usual absolute value for complex numbers.)
2. Retain the notation of #1. Show that R has exactly 6 units (1, -w², w, -1, w², -w). Show that (3) is the square of a prime ideal in R. Also show that all other ordinary prime numbers p either remain prime in R or else split into products of two primes of R, which are distinct even up to associates.
3. Not required, for extra credit only: Find necessary and sufficient conditions for the two cases of part (2). In other words, determine when a prime number p splits and when it remains inert in R. Interpret the result in terms of solutions of the Diophantine equation p = a² + b² - a b.

1. Use the structure theorem for finitely generated modules over a PID to give a structure theorem for pairs (A, T), where A is a finitely generated abelian group and T is an automorphism of A with T² = -1. (Hint: if one defines ix = Tx for x in A, then A becomes a finitely generated module over R = Z[i], the Gaussian integers.)
2. Use your answer to (1) to classify all pairs (A, T) up to isomorphism, where A is an abelian group of order 25 and T is an automorphism of A of order precisely 4. (First figure out the possiblilities for A, and for each one, figure out the possibilities for T² before figuring out the possibilities for T.)