Cyclic Groups

Question 1

Cyclic Group

Answer 1

a group s.t. G=⟨a⟩ for some a in G

Question 2

Generator of a Cyclic Group

Answer 2

a in G s.t. ⟨a⟩ = G

Question 3

Theorem 1

aⁱ = a^j criterion

Answer 3

Let a in G.
(1) Suppose |a|=∞. Then aⁱ=a^j iff i = j

(2) Suppose |a| = n. Then ⟨a⟩ = {e, a, a², … , a^n-1} and aⁱ=a^j iff i ≡ j mod n

(3) |a| = |⟨a⟩|

Question 4

Theorem 2

Computing ⟨a^k⟩ and |a^k| from a

Answer 4

Let k in N and a in G w/ finite order.

(1) ⟨a^k⟩=⟨a^gcd(|a|,k)⟩

(2) |a^k| = |a|/ gcd(|a|, k )

Question 5

Theorem 3

Classification of Subgroups of a Cyclic Group

Answer 5

Let G = ⟨a⟩ be a finite cyclic group.

(1) Every subgroup of G is cyclic (also true if |G| = infinity)

(2) The order of any subgroup of G divides |G|

(3) If k divides |G| then ⟨a^|G|/k⟩ is the unique subgroup of order k

Question 6

Euler phi function

Answer 6

• φ(n) = |U(n)|
• (Corollary 5) Suppose G is finite cyclic. If d divides |G|, then G has φ(d) elements of order d