Exercises
Let
a. Prove that
b. Find the kernel of
c. Prove or disprove that
d. Prove or disprove that
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Elements Of Modern Algebra
- Exercises In Section 3.3, the centralizer of an element a in the group G was shown to be the subgroup given by Ca=xGax=xa. Use the multiplication table constructed in Exercise 20 to find the centralizer Ca for each element a of the octic group D4. Construct a multiplication table for the octic group D4 described in Example 12 of this section.arrow_forwardFind the right regular representation of G as defined Exercise 11 for each of the following groups. a. G={ 1,i,1,i } from Example 1. b. The octic group D4={ e,,2,3,,,, }.arrow_forward23. Let be a group that has even order. Prove that there exists at least one element such that and . (Sec. ) Sec. 4.4, #30: 30. Let be an abelian group of order , where is odd. Use Lagrange’s Theorem to prove that contains exactly one element of order .arrow_forward
- 24. Let be a group and its center. Prove or disprove that if is in, then and are in.arrow_forward12. Find all homomorphic images of each group in Exercise of Section. 18. Let be the group of units as described in Exercise. For each value of, write out the elements of and construct a multiplication table for . a. b. c. d.arrow_forwardLet a and b be elements of a group G. Prove that G is abelian if and only if (ab)2=a2b2.arrow_forward
- 15. Assume that can be written as the direct sum , where is a cyclic group of order . Prove that has elements of order but no elements of order greater than Find the number of distinct elements of that have order .arrow_forward41. Prove or disprove that the set in Exercise is a group with respect to 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.arrow_forwardIn Exercises , is a normal subgroup of the group . Find the order of the quotient group . Write out the distinct elements of and construct a multiplication table for . 3. The quaternion group ; .arrow_forward
- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,