Section 1.3. Groups 19 Note that the set Z× of all nonzero integers is not a multiplicative group, for none of its elements (aside from ±1) has a multiplicative inverse in Z×. (ii) The set Z of all integers is an additive abelian group with a ∗ b = a + b, with identity 0, and with the inverse of an integer n being −n. Similarly, we can see that Q, R, and C are additive abelian groups. (iii) The circle group, S1 = {z ∈ C : |z| = 1}, is the group of all numbers of modulus 1 (the modulus of z = a + ib ∈ C is |z| = √complex a2 + b2) with binary operation multiplication of complex numbers. The set S1 is closed, for if |z| = 1 = |w|, then |zw| = 1. Complex multiplication is associative, the identity is 1 (which has modulus 1), and the inverse of any complex number z = a + ib of modulus 1 is its complex conjugate z = a − ib (which also has modulus 1). Thus, S1 is a group. (iv) For any positive integer n, let μn = {z ∈ C : zn = 1} be the set of all the nth roots of unity with binary operation multiplication of complex numbers. Now μn is an abelian group: the set μn is closed [if zn = 1 = wn, then (zw)n = znwn = 1] 1n = 1 multiplication is associative and commutative the inverse of any nth root of unity is its complex conjugate, which is also an nth root of unity. (v) The plane R2 is a group with operation vector addition that is, if α = (x, y) and α = (x , y ), then α + α = (x + x , y + y ). The identity is the origin O = (0, 0), and the inverse of (x, y) is (−x, −y). Example 1.18. Let X be a set. The Boolean group B(X) (named after the logician Boole) is the family of all the subsets of X equipped with addition given by symmetric difference A + B, where A + B = (A − B) ∪ (B − A) (recall that A − B = {x ∈ A : x / ∈ B}). Symmetric difference is pictured in Figure 1.3. It is plain that A + B = B + A, so that symmetric difference is A B Figure 1.3. Symmetric difference. A B C Figure 1.4. Associativity. commutative. The identity is ∅, the empty set, and the inverse of A is A itself, for

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