For an arbitrary set
Prove that
If
Want to see the full answer?
Check out a sample textbook solutionChapter 3 Solutions
Elements Of Modern Algebra
- Exercises 31. Let be a group with its center: . Prove that if is the only element of order in , then .arrow_forwardExercises 30. For an arbitrary positive integer, prove that any two cyclic groups of order are isomorphic.arrow_forwardExercises 35. Prove that any two groups of order are isomorphic.arrow_forward
- Find two groups of order 6 that are not isomorphic.arrow_forwardFor each a in the group G, define a mapping ta:GG by ta(x)=axa1. Prove that ta is an automorphism of G. Sec. 4.6,32 Let a be a fixed element of the group G. According to Exercise 20 of Section 3.5, the mapping ta:GG defined by ta(x)=axa1 is an automorphism of G. Each of these automorphisms ta is called an inner automorphism of G. Prove that the set Inn(G)=taaG forms a normal subgroup of the group of all automorphisms of G.arrow_forward43. Suppose that is a nonempty subset of a group . Prove that is a subgroup of if and only if for all and .arrow_forward
- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,