Cyclic Groups

Definition

• G is a cyclic group if G = <a> for some a in G.

Cyclic groups may be finite:

• In Z4, <1> = {1, 2, 3, 0} = Z4

• Z4 is cyclic.

• In Z5, <1> = {1,2,3,4,0} = Z5

• Z5 is cyclic

• In Zn, <1> = {1,2,3,…,(n-1),0} = Zn

• Zn is cyclic for n ≥ 1.

Cyclic groups may be infinite.

• In Z, <1> =

• In Z, <2> =

• In Q* <2> =

• In GL(2,R)

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1 1

0 1

⎡

⎣ ⎢

⎤

⎦ ⎥ =

1 n

0 1

⎡

⎣ ⎢

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⎦ ⎥ n ∈ Z

⎧ ⎨ ⎪

⎩ ⎪

⎫ ⎬ ⎪

⎭ ⎪

{ … -2, -1, 0, 1, 2, …}

{ … -4, -2, 0, 2, 4, …}{ … 1/4, 1/2, 1, 2, 4, …}

Quick Facts: Cyclic Groups

• Every cyclic group is Abelian.• Not every Abelian group is cyclic.• If H ≤ <a>, then H =<am> where m is the smallest

positive integer with am in H.• Every subgroup of a cyclic group is cyclic.

• Let G = <a> with |G|=n• The order of each subgroup of G divides n.• There is exactly one subgroup of G with order k,

namely, <an/k>.

• Cyclic groups are the building blocks of all Abelian groups.

• 5 10 15 20 25 30 35 40 45 50 55 60

5 25 50 10 35 60 20 45 5 30 55 15 40

10 50 35 20 5 55 40 25 10 60 45 30 15

15 10 20 30 40 50 60 5 15 25 35 45 55

20 35 5 40 10 45 15 50 20 55 25 60 30

25 60 55 50 45 40 35 30 25 20 15 10 5

30 20 40 60 15 35 55 10 30 50 5 25 45

35 45 25 5 50 30 10 55 35 15 60 40 20

40 5 10 15 20 25 30 35 40 45 50 55 60

45 30 60 25 55 20 50 15 45 10 40 5 35

50 55 45 35 25 15 5 60 50 40 30 20 10

55 15 30 45 60 10 25 40 55 5 20 35 50

60 40 15 55 30 5 45 20 60 35 10 50 25

Group G*mod 65

Group G*mod 65Reorderedas <50>

• 40 50 30 5 55 20 25 15 35 60 10 45

40 40 50 30 5 55 20 25 15 35 60 10 45

50 50 30 5 55 20 25 15 35 60 10 45 40

30 30 5 55 20 25 15 35 60 10 45 40 50

5 5 55 20 25 15 35 60 10 45 40 50 30

55 55 20 25 15 35 60 10 45 40 50 30 5

20 20 25 15 35 60 10 45 40 50 30 5 55

25 25 15 35 60 10 45 40 50 30 5 55 20

15 15 35 60 10 45 40 50 30 5 55 20 25

35 35 60 10 45 40 50 30 5 55 20 25 15

60 60 10 45 40 50 30 5 55 20 25 15 35

10 10 45 40 50 30 5 55 20 25 15 35 60

45 45 40 50 30 5 55 20 25 15 35 60 10

Group GListing onlypowers

• 0 1 2 3 4 5 6 7 8 9 10 11

0 0 1 2 3 4 5 6 7 8 9 10 11

1 1 2 3 4 5 6 7 8 9 10 11 0

2 2 3 4 5 6 7 8 9 10 11 0 1

3 3 4 5 6 7 8 9 10 11 0 1 2

4 4 5 6 7 8 9 10 11 0 1 2 3

5 5 6 7 8 9 10 11 0 1 2 3 4

6 6 7 8 9 10 11 0 1 2 3 4 5

7 7 8 9 10 11 0 1 2 3 4 5 6

8 8 9 10 11 0 1 2 3 4 5 6 7

9 9 10 11 0 1 2 3 4 5 6 7 8

10 10 11 0 1 2 3 4 5 6 7 8 9

11 11 0 1 2 3 4 5 6 7 8 9 10