If a is a group element of infinite order, what is a generator of the subgroup <am> <an>?

True or False Label each of the following statements as either true or false. In a Cayley table for a group, each element appears exactly once in each row.

Prove that Ca=Ca1, where Ca is the centralizer of a in the group G.

Find the order of each of the following elements in the multiplicative group of units . for for for for

Exercises 10. Find an isomorphism from the multiplicative group to the group with multiplication table in Figure . This group is known as the Klein four group. Figure Sec. 16. a. Prove that each of the following sets is a subgroup of , the general linear group of order over . Sec. 3. Let be the Klein four group with its multiplication table given in Figure . Figure Sec. 17. Show that a group of order either is cyclic or is isomorphic to the Klein four group . Sec. 16. Repeat Exercise with the quaternion group , the Klein four group , and defined by

True or False Label each of the following statements as either true or false. An element in a group may have more than one inverse.

Show that a group of order 4 either is cyclic or is isomorphic to the Klein four group e,a,b,ab=ba.

34. Suppose that and are subgroups of the group . Prove that is a subgroup of .

38. Let be the set of all matrices in that have the form with all three numbers , , and nonzero. Prove or disprove that is a group with respect to multiplication.

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