show that each of the following collections of numbers forms a group under addition (i) The even integers. (ii) All real numbers of the form a + bsqrt(2) where a,b E Z (iii) All real numbers of the form a + bsqrt(2) where a,b E Q (iV) All complex numbers of the form a + bi where a,b E Z
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show that each of the following collections of numbers forms a group under addition
(i) The even integers.
(ii) All real numbers of the form a + bsqrt(2) where a,b E Z
(iii) All real numbers of the form a + bsqrt(2) where a,b E Q
(iV) All
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- 9. Find all homomorphic images of the octic 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 .Prove that the group in Exercise is cyclic, with as a generator. Prove that for a fixed value of , the set of all th roots of forms a group with respect to multiplication.
- Exercises 27. Consider the additive groups , , and . Prove that is isomorphic to .3. Consider the additive groups of real numbers and complex numbers and define by . Prove that is a homomorphism and find ker . Is an epimorphism? Is a monomorphism?Prove part c of Theorem 3.4. Theorem 3.4: Properties of Group Elements Let G be a group with respect to a binary operation that is written as multiplication. The identity element e in G is unique. For each xG, the inverse x1 in G is unique. For each xG,(x1)1=x. Reverse order law: For any x and y in G, (xy)1=y1x1. Cancellation laws: If a,x, and y are in G, then either of the equations ax=ay or xa=ya implies that x=y.
- Exercises 9. Find an isomorphism from the multiplicative group of nonzero complex number to the multiplicative group and prove that . Sec. 15. Prove that each of the following subsets of is a subgroup of , the general linear group of order over . a.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.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.)
- 1.Prove part of Theorem . Theorem 3.4: Properties of Group Elements Let be a group with respect to a binary operation that is written as multiplication. The identity element in is unique. For each, the inverse in is unique. For each . Reverse order law: For any and in ,. Cancellation laws: If and are in , then either of the equations or implies that .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.