# Extra Exercises

• Suppose that G is a group and that P is a partition of G (so P is a collection of pairwise disjoint non-empty subsets of G where the union of the collection is G). The product of two subsets of G is a subset of G (see the start of 1.3). Suppose that this operation renders P a group. Prove that if the identity element of G belongs to N which is one of the sets comprising P, then N is a normal subgroup of G, and P is isomorphic to G/N.
• Exhibit a specific group G and a subset P of the power set of G such that the union of all the sets in P is G and the collection P forms a group under multiplication of sets and (finally) P is not a partition of G (a question suggested by Gunnar Traustason).
• Let G be a group and suppose that g in G has finite order n. Show that if p_1, ..., p_m are the distinct prime divisors of n, then g is the product of pairwise commuting elements g_1, ..., g_m of G where the order of g_j is a power of p_j.
• From the previous question, deduce the following result about the rational numbers: Suppose that x, n are integers and n is not 0. Suppose that p_1, ..., p_m are the distinct prime divisors of n. It follows that x/n is the sum of m rational numbers a_1, ..., a_m where each a_j in lowest terms has denominator which is a power of p_j. In what sense is this expression unique?

Here are some problem sheets I have used when teaching an undergraduate course based on the first half of the book. I do not supply answers, so that academics can use them as problems for homework. These were used in 1999.

 Sheet 1 adobe pdf plain text dvi postscript LaTeX2e source. Sheet 2 adobe pdf plain text dvi postscript LaTeX2e source. Sheet 3 adobe pdf plain text dvi postscript LaTeX2e source. Sheet 4 adobe pdf plain text dvi postscript LaTeX2e source. Sheet 5 adobe pdf plain text dvi postscript LaTeX2e source. Sheet 6 adobe pdf plain text dvi postscript LaTeX2e source. Sheet 7 adobe pdf plain text dvi postscript LaTeX2e source Sheet 8 adobe pdf plain text dvi postscript LaTeX2e source Sheet 9 adobe pdf plain text dvi postscript LaTeX2e source