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To reinforce Angelo's example, it's worthwhile to point out the broader setting for this kind of question: the study of centralizers and connectedness properties in a semisimple (or more generally reductive) algebraic group. An older but very useful source is part II of the extensive notes by T.A. Springer and R. Steinberg on conjugacy classes, part of an IAS seminar (Lect. Notes in Math. 131, Springer, 1970). A crucial question is whether a given connected semisimple group is simply connected or not; this shows up in the standard example where the adjoint group $\mathrm{PGL}$ fails to be simply connected. Here you have the deep theorem: If $G$ is a connected, simply connected algebraic group over an algebraically closed field, then all centralizers of semisimple elements are connected. (It's elementary on the other hand to prove that all centralizers in a general linear group are connected.) The role of the characteristic of the field is also discussed in depth by Springer and Steinberg, as well as the role of "torsion primes" (treated mpre fully in Steinberg's 1975 Advances paper).

Some of the results are written up in later textbooks and in the first two chapters of my 1990 AMS book Conjugacy Classes in Semisimple Algebraic Groups (with the relevant example for the question here given in 1.12).

ADDED: To answer the added question, in any connected algebraic group it's true that an arbitrary semisimple element and hence the cyclic subgroup it generates lies in some maximal torus. This is part of the standard development of Borel-Chevalley structure theory (see for example Section 22.3 of my book Linear Algebraic Groups), though it does take a while to get that far into the theory.

To reinforce Angelo's example, it's worthwhile to point out the broader setting for this kind of question: the study of centralizers and connectedness properties in a semisimple (or more generally reductive) algebraic group. An older but very useful source is part II of the extensive notes by T.A. Springer and R. Steinberg on conjugacy classes, part of an IAS seminar (Lect. Notes in Math. 131, Springer, 1970). A crucial question is whether a given connected semisimple group is simply connected or not; this shows up in the standard example where the adjoint group $\mathrm{PGL}$ fails to be simply connected. Here you have the deep theorem: If $G$ is a connected, simply connected algebraic group over an algebraically closed field, then all centralizers of semisimple elements are connected. (It's elementary on the other hand to prove that all centralizers in a general linear group are connected.) The role of the characteristic of the field is also discussed in depth by Springer and Steinberg, as well as the role of "torsion primes" (treated more fully in Steinberg, Torsion in reductive groups, Advances in Math 1975).

Some of the results are written up in later textbooks and in the first two chapters of my 1990 AMS book Conjugacy Classes in Semisimple Algebraic Groups (with the relevant example for the question here given in 1.12).

ADDED: To answer the added question, in any connected algebraic group it's true that an arbitrary semisimple element and hence the cyclic subgroup it generates lies in some maximal torus. This is part of the standard development of Borel-Chevalley structure theory (see for example Section 22.3 of my book Linear Algebraic Groups), though it does take a while to get that far into the theory.

To reinforce Angelo's example, it's worthwhile to point out the broader setting for this kind of question: the study of centralizers and connectedness properties in a semisimple (or more generally reductive) algebraic group. An older but very useful source is part II of the extensive notes by T.A. Springer and R. Steinberg on conjugacy classes, part of an IAS seminar (Lect. Notes in Math. 131, Springer, 1970). A crucial question is whether a given connected semisimple group is simply connected or not; this shows up in the standard example where the adjoint group $\mathrm{PGL}$ fails to be simply connected. Here you have the deep theorem: If $G$ is a connected, simply connected algebraic group over an algebraically closed field, then all centralizers of semisimple elements are connected. (It's elementary on the other hand to prove that all centralizers in a general linear group are connected.) The role of the characteristic of the field is also discussed in depth by Springer and Steinberg, in relation to "torsion primes".

Some of the results are written up in later textbooks and in the first two chapters of my 1990 AMS book Conjugacy Classes in Semisimple Algebraic Groups (with the relevant example for the question here given in 1.12).

ADDED: To answer the added question, in any connected algebraic group it's true that an arbitrary semisimple element and hence the cyclic subgroup it generates lies in some maximal torus. This is part of the standard development of Borel-Chevalley structure theory (see for example Section 22.3 of my book Linear Algebraic Groups), though it does take a while to get that far into the theory.

To reinforce Angelo's example, it's worthwhile to point out the broader setting for this kind of question: the study of centralizers and connectedness properties in a semisimple (or more generally reductive) algebraic group. An older but very useful source is part II of the extensive notes by T.A. Springer and R. Steinberg on conjugacy classes, part of an IAS seminar (Lect. Notes in Math. 131, Springer, 1970). A crucial question is whether a given connected semisimple group is simply connected or not; this shows up in the standard example where the adjoint group $\mathrm{PGL}$ fails to be simply connected. Here you have the deep theorem: If $G$ is a connected, simply connected algebraic group over an algebraically closed field, then all centralizers of semisimple elements are connected. (It's elementary on the other hand to prove that all centralizers in a general linear group are connected.) The role of the characteristic of the field is also discussed in depth by Springer and Steinberg, as well as the role of "torsion primes" (treated mpre fully in Steinberg's 1975 Advances paper).

Some of the results are written up in later textbooks and in the first two chapters of my 1990 AMS book Conjugacy Classes in Semisimple Algebraic Groups (with the relevant example for the question here given in 1.12).

ADDED: To answer the added question, in any connected algebraic group it's true that an arbitrary semisimple element and hence the cyclic subgroup it generates lies in some maximal torus. This is part of the standard development of Borel-Chevalley structure theory (see for example Section 22.3 of my book Linear Algebraic Groups), though it does take a while to get that far into the theory.

To reinforce Angelo's example, it's worthwhile to point out the broader setting for this kind of question: the study of centralizers and connectedness properties in a semisimple (or more generally reductive) algebraic group. An older but very useful source is part II of the extensive notes by T.A. Springer and R. Steinberg on conjugacy classes, part of an IAS seminar (Lect. Notes in Math. 131, Springer, 1970). A crucial question is whether a given connected semisimple group is simply connected or not; this shows up in the standard example where the adjoint group $\mathrm{PGL}$ fails to be simply connected. Here you have the deep theorem: If $G$ is a connected, simply connected algebraic group over an algebraically closed field, then all centralizers of semisimple elements are connected. (It's elementary on the other hand to prove that all centralizers in a general linear group are connected.) The role of the characteristic of the field is also discussed in depth by Springer and Steinberg, in relation to "torsion primes".

Some of the results are written up in later textbooks and in the first two chapters of my 1990 AMS book Conjugacy Classes in Semisimple Algebraic Groups (with the relevant example for the question here given in 1.12).

To reinforce Angelo's example, it's worthwhile to point out the broader setting for this kind of question: the study of centralizers and connectedness properties in a semisimple (or more generally reductive) algebraic group. An older but very useful source is part II of the extensive notes by T.A. Springer and R. Steinberg on conjugacy classes, part of an IAS seminar (Lect. Notes in Math. 131, Springer, 1970). A crucial question is whether a given connected semisimple group is simply connected or not; this shows up in the standard example where the adjoint group $\mathrm{PGL}$ fails to be simply connected. Here you have the deep theorem: If $G$ is a connected, simply connected algebraic group over an algebraically closed field, then all centralizers of semisimple elements are connected. (It's elementary on the other hand to prove that all centralizers in a general linear group are connected.) The role of the characteristic of the field is also discussed in depth by Springer and Steinberg, in relation to "torsion primes".

Some of the results are written up in later textbooks and in the first two chapters of my 1990 AMS book Conjugacy Classes in Semisimple Algebraic Groups (with the relevant example for the question here given in 1.12).

ADDED: To answer the added question, in any connected algebraic group it's true that an arbitrary semisimple element and hence the cyclic subgroup it generates lies in some maximal torus. This is part of the standard development of Borel-Chevalley structure theory (see for example Section 22.3 of my book Linear Algebraic Groups), though it does take a while to get that far into the theory.