S2, C4: Cyclic Groups

Question 1

Definitions 4.1. Cyclic Groups

Answer 1

We let (g) denote the set {g^n: n ∈ Z} of all powers of g. It is easy to see (with the subgroup criterion) that (g) is a subgroup of G called the cyclic subgroup generated by g.
A group G is said to be cyclic if G=(g)  for some g∈G; that is, if G consists of the powers of one of its elements g.

Question 2

Definition 4.9. Order of an element of a Group

Answer 2

For an element of a group G, we define the order of g to be the least positive integer n such that g^n=e, if such an integer exists, and to be infinite if no such integer exists.

Question 3

Let G be a group and let gϵG have finite order n. Then…

Answer 3

1. g^m = g^r, where r is the remainder on division of m by n
2. g^m = e iff m is a multiple of n
3. the order of the group (g) is the same as the order of g.

Question 4

Let G be a finite group of order n. Then G is cyclic iff…

Answer 4

G has an element of order n.

Question 5

Which cyclic groups are abelian?

Answer 5

All.

Question 6

Let g be a a group element of finite order n. Then the order of g^m is…

Answer 6

… l/m where l is the lcm of m and n.

If m is a factor of n then the order of g^m is n/m.

Question 7

Subgroups of cyclic groups are…

Answer 7

…cyclic.

Question 8

Isomorphisms between cyclic groups.

Answer 8

Let G= and H= be cyclic groups of the same order. Then there is a bijection f: G -> H given by the rule f(g^i) = h^i for any integer i. f is an isomorphism of groups.

Cyclic groups of the same order are isomorphic.