This chapter introduces a simple method of constructing a new group from two given groups, which is useful both for construction and for analysis. The principal use that we will make of this construction, in later chapters, is the construction of groups of vectors. Qn 11 gives the key theorem for decomposition.

Concurrent reading: Fraleigh, section 8.

1. 1 If (A, ·) is the cyclic group C2 = {e, a), the cartesian product A × A consists of the four elements (e, e), (e, a), (a, e) and (a, a). Make a table to exhibit all possible products of these elements under the operation defined by

2. (b1, c1)(b2, c2) = (b1 · b2, · c2).

3. Is the table that of a group isomorphic to D2? The group constructed in this way is called the direct product C2 × C2.

4. 2 By analogy with qn 1, construct the direct product A × B, where A is the cyclic group C2 = {e, a} and B is the cyclic group C3 = {e, b, b2}. Find the order of the element (a, b) in C2 × C3 and determine whether C2 × C3 is cyclic.

5. 3 As in qn 2, construct the direct product C2 × C5 and show that this is a cyclic group.

6. 4 Construct the direct product C2 × C4 and find the order of each element in this group. Is it a cyclic group?

7. 5 […]