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ISBN 978 1 4730 2036 8 2.1 Contents Contents 1 Direct products 5 1.1 The external direct product of two groups 5 1.2 Internal direct products 8 2 Cyclic groups 18 2.1 The classification of cyclic groups 19 2.2 Subgroups and quotient groups of cyclic groups 20 2.3 Direct products of cyclic groups 24 3 Describing groups 32 3.1 Generators and relations 32 3.2 Dicyclic groups 38 Solutions and comments on 42 exercises Index 54 1 Direct products 1 Direct products This section describes how to construct a group called the direct product of two given groups, and then describes certain conditions under which a group can be regarded as the direct product of its subgroups. 1.1 The external direct product of two groups In this subsection, we introduce a method of constructing a new group from two existing ones: their external direct product. This method of construction will enable us to use existing groups as building blocks for larger ones. In turn, this will enable us to describe some groups in terms of smaller and possibly less complicated groups. This construction is, therefore, a key tool in our progress towards classifying groups. First we need the notion of the Cartesian product of two sets: if X and Y are two sets, then the Cartesian product X × Y is the set of all ordered pairs (x,y) with x ∈ X and y ∈ Y . For example, if X = {1, 2, 3} and Y = {a, b} then X × Y = {(1, a), (1, b), (2, a), (2, b), (3, a), (3, b)}. When X = Y , the Cartesian product X × Y = X × X is also denoted by 2 X . More generally, the Cartesian product of n sets X1, X2,...,Xn is the set of all ordered n-tuples (x1,x2,...,xn) with xi ∈ Xi for all i = 1,...,n. In order to define the direct product of two groups, we need to specify the underlying set and a suitable binary operation. To understand how this is done, it will be useful to examine the familiar example of addition of vectors in the real plane R × R represented by the usual coordinate system. Example 1.1 Addition of vectors in R × R We know that the real numbers with addition are a group, which we denoted by (R, +) in Chapter 5. Each point in the plane is represented by an ordered pair (x,y) of real numbers. In other words, the coordinates of points in the plane are elements of the Cartesian product R × R = {(x,y) : x ∈ R, y ∈ R} . The familiar operation of vector addition on R × R is defined as follows: (x1,y1) + (x2,y2) = (x1 + x2,y1 + y2). Thus, vector addition is defined component by component in terms of addition in R. It turns out that R × R with addition defined as in Example 1.1 above is a group. 5 Towards classification Exercise 1.1 Prove that the set R × R with the operation of vector addition is a group. We can generalise the definitions of both the real plane as a Cartesian product and component-wise addition of real numbers. Given two groups (G, ◦) and (H, ∗), we define the underlying set of a new In Example 1.1, G = H = R. group to be the Cartesian product of the sets G and H; that is, the set G × H = {(g, h): g ∈ G, h ∈ H}. As with R × R, we can define an operation on this set by applying the original operations to the separate components. Generalising requires a little care because the operations for the two component groups may be different. As the first components come from G, with operation ◦, and the second components come from H, with operation ∗, the general definition In Example 1.1, • = ◦ = ∗ = +. of the new operation (which we call •) is (g1, h1) • (g2, h2) = (g1 ◦ g2, h1 ∗ h2). Definition 1.2 Direct product of two groups Let (G, ◦) and(H, ∗) be two groups. Their direct product (G × H, •) is the group defined as follows. Set The underlying set of the group is G × H, the Cartesian product of the underlying sets G and H; that is, G × H = {(g, h): g ∈ G, h ∈ H} . Operation If (g1, h1) and (g2, h2) are elements of G × H then (g1, h1) • (g2, h2) = (g1 ◦ g2, h1 ∗ h2). This definition does indeed produce a group. Before we prove this, the following exercise will give you some ‘hands-on’ experience of direct products of groups. Exercise 1.2 Write out the Cayley table of the group Z2 × Z3. Use + for the new operation. 6 1 Direct products We are now ready to prove that the direct product of any two groups is indeed a group. The proof for the general case is similar to the solution to Exercise 1.1. To verify that (G × H, •) is a group, we must check the four group axioms. To check that the above definition of • produces a closed binary operation on G × H (that is, it satisfies the closure axiom), we observe the following. Let (g1, h1) and (g2, h2) be any two elements of G × H. By the definition of G × H, we know that g1 and g2 are elements of G and that h1 and h2 are elements of H. Since G is a group, it is closed under the operation ◦, therefore g1 ◦ g2 is in G. Since H is a group, it is closed under the operation ∗, so h1 ∗ h2 is in H. Therefore, by the definition of •, we have (g1, h1) • (g2, h2) = (g1 ◦ g2, h1 ∗ h2) ∈ G × H. The following exercise asks you to check the remaining group axioms for (G × H, •). Exercise 1.3 (a) Prove that • is associative (that is, check the associativity axiom). (b) Let the identity of G be eG and the identity of H be eH . Prove that (eG, eH ) is the identity of G × H (that is, check the identity axiom). (c) Let (g, h) be an element of G × H, where g has inverse g−1 in G and h has inverse h−1 in H. Prove that (g−1, h−1) is the inverse of (g, h) in G × H (that is, check the inverses axiom). So far we have carefully distinguished the binary operations of the groups involved in the direct product construction. Now that we have shown that the direct product of two groups is a group, we revert to using either juxtaposition for the operations, or + when both the operations are addition. Some texts refer to the direct sum of two groups, rather than direct product, when the groups are abelian and additive notation is used.