ch 4 to 5: cyclic groups and permutation groups

Theorem 4.1 Criterion for aⁱ = a^m

Let G be a group, and let a belong to G. If a has infinite order, then aⁱ = a^m if and only if

i = m

Theorem 4.1 Criterion for aⁱ = a^m

Let G be a group, and let a belong to G. If a has finite order, say, n, then <a> = {e, a, a² , … , a^n-1 } and aⁱ = a^m if and only if

n divides i - m

Corollary 1

For any group element a, |a| =

|<a>|

Corollary 2

If any group element a, a^k = e if and only if

|a| divides k

Corollary 3

For any group element a, a^k = e if and only if

k is a multiple of |a|

Corollary 4

If a and b belong to a finite group and ab = ba, then

|ab| divides |a| |b|

Theorem 4.2

Let a be an element of order n in a group and let k be a positive integer. Then

<a^k> = <a ^{gcd(n, k)}> and |a^k| = n/gcd(n, k)

Corollary 1

In a finite cyclic group, the order of an element divides the

order of the group

Corollary 2

Let |a| = n. Then <aⁱ> = <a^m> if and only if

gcd(n, i) = gd(n, m)

Corollary 2

Let |a| = n. |aⁱ| = |a^m| if and only if

gcd(n, i) = gcd(n, m)

Corollary 3

Let |a| = n. Then <a> = <a^m> if and only if

gcd(n, m) = 1

Corollary 3

Let |a| = n. |a| = |<a^m>| if and only if

gcd(n, m) = 1

Corollary 4

An integer k in Z_n is a generator of Z_n if and only if

gcd(n, k) = 1

Theorem 4.3 - Fundamental Theorem of Cyclic Groups pt 1

Every subgroup of a cyclic group is

cyclic

Theorem 4.3 - Fundamental Theorem of Cyclic Groups pt 2

MOreover, if |<a>| = n, then the order of any subgroup of <a> is a

divisor of n

Theorem 4.3 - Fundamental Theorem of Cyclic Groups pt 3

for each positive divisor k of n, the group <a> has exactly

one subgroup of order k - namely <a^n/k>

Corollary Subgroups of Z_n

For each positive divisor k of n, the set <n/k> is

the unique subgroup of Z_n of order k; moreover, these are only subgroups of Z_n

Theorem 4.4

If d is a positive divisor of n, the number of elements of order d in a cyclic group of order n is

Φ(d)

(Φ(p) = (p-1) for p is a prime and Φ(n) = n (p1 -1/p1) … (pk -1/ pk)

Corollary 4.1

In a finite group, the number of elements of order d is a

multiple of Φ(d)

A permutation of a set A is a

function from A to A that is both one-to-one and onto

A permutation group of a set A is a

set of permutations of A that forms a group under function composition

Let A = {1, 2, …, n}. S_n is the

set of all-permutation of A

The order of S_n is

n!

S_n is non-Abelian when

n ≥ 3

Theorem 5.1

Every permutation of a finite set can be written as a

cycle or a product of disjoint cycles

Theorem 5.2

If the pair of cycles α = (a₁ , a₂ , …., a_m ) and β = (b₁, b₂ ,…, b_n ) have no entries in common, then

αβ = βα

Theorem 5.3

The order of a permutation of a finite set written in disjoint cycle form is the

least common multiple of the lengths of the cycles

Theorem 5.4

Every permutation in S_n , n > 1, is a

product of 2-cycles

Lemma

If ε = β₁ β₂ … β_r , where the β’s are 2-cycles, then

r is even

Theorem 5.5

If a permutation α can be expressed as a product of an even (odd) number of 2-cycles, then

every decomposition of α into a product of a 2-cycles must an even (odd) number 2-cycles α = β₁ β₂ … β_r and α = γ₁ γ₂ … γ_s , where the β’s and γ’s are 2-cycles, then r and s are both even or both odd

An even permutation

is a permutation that can be expressed as a product of an even number of 2-cycles

An odd permutation

is a permutation that can expressed as a product of an odd number of 2-cycles

Theorem 5.6

The set of even permutations in S_n forms a

subgroup of S_n

A_n - The alternating group of degree n is

the group of even permutations of n symbols

Theorem 5.7

For n > 1, A_n has the order

n!/2