Determine the behaviour of the set of 4 elements {1,-1,i,-i} under the ordinary multiplication of complex numbers. Show that it is a group.
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- In Exercises 114, decide whether each of the given sets is a group with respect to the indicated operation. If it is not a group, state a condition in Definition 3.1 that fails to hold. The set of all complex numbers x that have absolute value 1, with operation addition. Recall that the absolute value of a complex number x written in the form x=a+bi, with a and b real, is given by | x |=| a+bi |=a2+b2Exercises In Exercises, decide whether each of the given sets is a group with respect to the indicated operation. If it is not a group, state a condition in Definition that fails to hold. 9. The set of all complex numbers that have absolute value , with operation multiplication. Recall that the absolute value of a complex number written in the form, with and real, is given by.9. Find all homomorphic images of the octic group.
- Find all homomorphic images of the quaternion group.Find all subgroups of the quaternion group.Exercises 16. Assume that the nonzero complex numbers form a group with respect to multiplication. If and are real numbers and , the conjugate of the complex number is defined to be . With this notation, let be defined by for all in . Prove that is an automorphism of .
- Exercises In Exercises, decide whether each of the given sets is a group with respect to the indicated operation. If it is not a group, state a condition in Definition that fails to hold. 8. For a fixed positive integer, the set of all complex numbers such that (that is, the set of all roots of), with operation multiplication.Use mathematical induction to prove that if a1,a2,...,an are elements of a group G, then (a1a2...an)1=an1an11...a21a11. (This is the general form of the reverse order law for inverses.)19. a. Show that is isomorphic to , where the group operation in each of , and is addition. b. Show that is isomorphic to , where all group operations are addition.
- 39. Let be the set of all matrices in that have the form for arbitrary real numbers , , and . Prove or disprove that is a group with respect to multiplication.16. Suppose that is an abelian group with respect to addition, with identity element Define a multiplication in by for all . Show that forms a ring with respect to these operations.Let a,b,c, and d be elements of a group G. Find an expression for (abcd)1 in terms of a1,b1,c1, and d1.