3:02 ← abstract algebra by fral... 73% group. Smit 210/20} is a 1 1/2(0), 11 y 2 ) 1s une Canonica map, then by Theorem 15.16, y¯¹[Z(G/Z(G))] is a normal subgroup Z₁(G) of G. We can then form the factor group G/Z₁(G) and find its center, take (1/1)-1 of it to get Z2(G) and so on. 35.20 Definition The series 35.21 Example {e} ≤ Z(G) ≤ Z₁(G) ≤ Z₂(G) ≤... described in the preceding discussion is the ascending central series of the group G The center of S3 is just the identity {00). Thus the ascending central series of S3 is {po} ≤ {0} ≤ {po} ≤ …….. The center of the group D4 of symmetries of the square in Example 8.10 is {po, P2} (Do you remember that we said that this group would give us nice examples of many things we discussed?) Since D4/{po, p2} is of order 4 and hence abelian, its center is al of D4/{00, P2}. Thus the ascending central series of D4 is {00} ≤ {P0, P2} ≤ D4 ≤ D4 ≤ D4 ≤ …….. Section 35 Exercises 319 EXERCISES 35 Computations In Exercises 1 through 5, give isomorphic refinements of the two series. 1. {0} < 10Z < Z and {0} < 25Z < Z 2. {0} < 60Z < 20Z < Z and {0} < 245Z < 49Z < Z 3. {0} < < (3) < Z24 and {0} < (8) < Z24 4. {0} < (18) < (3) < Z72 and {0} < (24) < (12) < Z72 5. {(0,0)} < (60Z) × Z < (10Z) × Z < Z × Z and {(0, 0)} < Z × (80Z) < Z × (20Z) < Z × Z 6. Find all composition series of Z60 and show that they are isomorphic. 7. Find all composition series of Z48 and show that they are isomorphic. 8. Find all composition series of Z5 × Z5. 9. Find all composition series of S3 × Z2. 10. Find all composition series of Z2 × Z5 × Z7. 11. Find the center of S3 × Z4. 12. Find the center of S3 × D4. 13. Find the ascending central series of S3 × Z4. 14. Find the ascending central series of S3 × D4. Concepts In Exercises 15 and 16, correct the definition of the italicized term without reference to the text, if correction is needed, so that it is in a form acceptable for publication. 15. A composition series of a group G is a finite sequence {e} = H0 < H₁ < H₂ < · · · < H₁−1 < H₂ = G of subgroups of G such that H₁ is a maximal normal subgroup of H+1 for i = 0, 1, 2, ..., n − 1. 16. A solvable group is one that has a composition series of abelian groups. 17. Mark each of the following true or false. a. Every normal series is also subnormal. b. Every subnormal series is also normal. c. Every principal series is a composition series. d. Every composition series is a principal series. e. Every abelian group has exactly one composition series. f. Every finite group has a composition series. g. A group is solvable if and only if it has a composition series with simple factor groups. h. S7 is a solvable group. i. The Jordan-Hölder theorem has some similarity with the Fundamental Theor which states that every positive integer greater than 1 can be factored into a uniquely up to order. j. Every finite group of prime order is solvable. tic, es 2/