Find an isomorphism from the multiplicative group
to the group
Figure
Sec.
a. Prove that each of the following sets
Sec.
Let
Figure
Sec.
Show that a group of order
Sec.
Repeat Exercise with the quaternion group
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Chapter 3 Solutions
Elements Of Modern Algebra
- Exercises 30. For an arbitrary positive integer, prove that any two cyclic groups of order are isomorphic.arrow_forwardExercises 8. Find an isomorphism from the group in Example of this section to the multiplicative group . Sec. 16. Prove that each of the following sets is a subgroup of , the general linear group of order over .arrow_forwardIf G is a cyclic group, prove that the equation x2=e has at most two distinct solutions in G.arrow_forward
- Let A={ a,b,c }. Prove or disprove that P(A) is a group with respect to the operation of union. (Sec. 1.1,7c)arrow_forward6. For each of the following values of , describe all the abelian groups of order , up to isomorphism. b. c. d. e. f.arrow_forward9. Suppose that and are subgroups of the abelian group such that . Prove that .arrow_forward
- Write 20 as the direct sum of two of its nontrivial subgroups.arrow_forward45. Let . Prove or disprove that is a group with respect to the operation of intersection. (Sec. )arrow_forwardProve 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.arrow_forward
- Let G be an abelian group of order 2n, where n is odd. Use Lagranges Theorem to prove that G contains exactly one element of order 2.arrow_forward39. 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.arrow_forward13. Assume that are subgroups of the abelian group . Prove that if and only if is generated byarrow_forward
- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,