Show that the elements of S9 which send the numbers 2, 5, 7 among themselves form a subgroup of S9. what is the order of this subgroup?

11. Find all normal subgroups of the alternating group .

Find a subset of Z that is closed under addition but is not subgroup of the additive group Z.

Find two groups of order 6 that are not isomorphic.

Find subgroups H and K of the group S(A) in example 3 of section 3.1 such that HK is not a subgroup of S(A). From Example 3 of section 3.1: A=1,2,3 and S(A) is a set of all permutations defined on A.

Decide whether each of the following sets is a subgroup of G={ 1,1,i,i } under multiplication. If a set is not a subgroup, give a reason why is not. a. { 1,1 } b. { 1,i } c. { i,i } d. { 1,i }

19. Prove that each of the following subsets of is a subgroup of . a. b.

40. Find subgroups and of the group in example of the section such that the set defined in Exercise is not a subgroup of . From Example of section : andis a set of all permutations defined on . defined in Exercise :

12. 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.