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Unformatted text preview: 94 2. BASIC THEORY OF GROUPS Example 2.2.3. In any group G , G itself and f e g are subgroups. Example 2.2.4. The set of all complex numbers of modulus (absolute value) equal to 1 is a subgroup of the group of all nonzero complex num bers, with multiplication as the group operation. See Appendix D . Proof. For any nonzero complex numbers a and b , j ab j D j a jj b j , and j a 1 j D j a j 1 . It follows that the set of complex number of modulus 1 is closed under multiplication and under inverses. n Example 2.2.5. In the group of symmetries of the square, the subset f e;r;r 2 ;r 3 g is a subgroup. Also, the subset f e;r 2 ;a;b g is a subgroup; the latter subgroup is isomorphic to the symmetry group of the rectangle, since each nonidentity element has square equal to the identity, and the product of any two nonidentity elements is the third. Example 2.2.6. In the permutation group S 4 , the set of permutations satisfying .4/ D 4 is a subgroup. This subgroup, since it permutes the numbers...
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 Fall '08
 EVERAGE
 Algebra, Complex Numbers

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