Groups 174 What's the most difficult aspect of your life as a mathematician, Diane Maclagan, an assistant professor at Rutgers, was asked. "Trying to prove theorems," she said. And the most fun? "Trying to prove theorems." Exercises 1. Prove that the external direct product of any finite number of sa group. (This exercise is referred to in this chapter.) 2. Show that Z, Z, Z, has seven subgroups of order 2 3. Let G be a group with identity e and let H be a group with iden- tity e Prove that G is isomorphic to Ge} and that H is iso morphic to (eg) H. 4. Show that G H is Abelian if and only if G and H are Abelian State the general case. 5. Prove or disprove that Z Z is a cyclic group. 6. Prove, by comparing orders of elements, that Zg Z, is not iso- morphic to Z, Z. 7. Prove that G, G, is isomorphic to G, G. State the general groups 2 8 4 case. 8. Is Z,Z, isomorphic to Z2? Why? 9. Is Z,z, isomorphic to Z,5? Why? 10. How many elements of order 9 does Z Z, have? (Do not do this exercise by brute force.) 11. How many elements of order 4 does Z, Z, have? (Do not do this by examining each element.) Explain why Z Z number of elements of order 4 as does Zs00000 Z400000 ize to the case Z Z. 29 30 has the same General- 31 32 12. Give examples of four groups of order 12, no two of which are isomorphic. Give reasons why no two are isomorphic. 13. For each integer n > 1, give examples of two nonisomorphic 14. The dihedral group D, of order 2n (n 3) has a subgroup of n ro- morphic to the external direct product of two such groups. of order n2. groups 33. tations and a subgroup of order 2. Explain why D, cannot be iso- 15. Prove that the group of complex numbers under addition is iso- 16. Suppose that G, G2 and H, H2. Prove that G, H G 34. 35. morphic to R R. 36. H,. State the general case. 17. If GH is cyclic, prove that G and H are 37. case. 18. In Z4 Z 38. cyclic. State the genera find two subgroups of order 12. 30
Percentage
A percentage is a number indicated as a fraction of 100. It is a dimensionless number often expressed using the symbol %.
Algebraic Expressions
In mathematics, an algebraic expression consists of constant(s), variable(s), and mathematical operators. It is made up of terms.
Numbers
Numbers are some measures used for counting. They can be compared one with another to know its position in the number line and determine which one is greater or lesser than the other.
Subtraction
Before we begin to understand the subtraction of algebraic expressions, we need to list out a few things that form the basis of algebra.
Addition
Before we begin to understand the addition of algebraic expressions, we need to list out a few things that form the basis of algebra.
8
Trending now
This is a popular solution!
Step by step
Solved in 4 steps with 2 images