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- StudyBlue
- California
- Mills College
- Mathematics
- Mathematics 132
- Berard
- Ch. 7: Groups

Lucy B.

Symmetry

a rearrangement of a shape the preserves distances, angles, etc.

Group

A group is a nonempty set G with a binary operation * that satisfies

- Closure under mult
- Associativity
- Identity Element
- Multiplicative inverses

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Abelian

if a*b = b*a for all a, b ∈ G, we say G is abelian.

Order

The number of elements in the group G. Also lGl

Dihedral Group of Degree n (D_{n})

All symmetries of a polygon of n sides

Symmetric group of n symbols (S_{n})

All permutations of {1, 2, ..... n}

If (R, +, •) Is a ring, then...

(R, +) is a group. (R, •) is never a group.

R^{*}

The set of all nonzero elements of a ring R.

If F is a field, then...

(F*,•) is a group.

GL_{n}(IR)

General Linear Group: The group of all invertible n x n matrices with entries from IR.

Suppose we have (G, *), (H, ™) and these are groups. Then, (G x H, ♣) is a group where?

(g, h) ♣ (g', h') = (g*g', h™h')

- G, H abelian means that GxH is abelian
- If G, H have finite order, GxH = order (G) x order (H)

If G is a group and a, b, c exist in G, then...(3 properties)

- G has a unique identity
- Cancellation holds in G
- ab = ac ⇒ b = c
- ba = ca ⇒ b = c
- Each element in G has a unique inverse.

G = gp, a, b ∈ G. Then, ...

- (ab)
^{-1}=(b^{-1}a^{-1}) - (a
^{-1})^{-1}=a

e

identity

order of an element a ∈ G

The smallest positive (>0) integer n s.t. a^{n}= e

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infinite order of a ∈ G

If a^{n} ≠ e for any positive integer, we say a has infinite order.

if H is closed under the operation in G, then...

H is a subgroup of G.

Center of a group

the center of a group G is:

Z(G) = {a∈ g l ag = ga for every g ∈ G}

A nonempty subset H of a group G is a subgroup of G if... (2 criteria)

- a, b ∈ H ⇒ ab ∈ H
- a ∈ H ⇒ a
^{-1}∈ H

Subgroup

H is a subgroup of a group G if H is a subset of G and H is a group under the operation of G.

Proper Subgroup

If H <G and H =/ e, then H is a proper subgroup.

Cyclic subgroup generated by a

the set of all powers of a.

<a> = {a^{n}l n ∈ Z}

If using additive notation:

<a> = {na l n ∈ Z}

cyclic group

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

<S>

Let S be a nonempty subset of a group G.

<S> = subset of all possible products, in every order, of all elements of S and their inverses.

nth roots of unity

the group of nth root of unity is the set of all solutions to x^{n}=1.

Isomorphism

(G, *), (H, ♠) are groups. Then G is isomorphic to H if there exists a bijective map Φ: G→ H such that the operation is preserved. Φ (a * b) = Φ(a) ♠ Φ (b).

automorphism

an isomorphism f: G → G

inner automorphism

Fix c in G. The inner automorphism of G induced by C is

f: G → G

f(g) = c^{-1}g c

Homomorphism

f: (G, *) → (H, ♠) is a homomorphism if f (a*b) = f(a) ♠ f(b) for all a, b in G.

Let Φ: G→H be an iso of 2 groups. Then...(6 things)

- Φ
^{-1}is an iso - Order of G = Order of H
- If G is abelian, H is abelian
- If G is Cyclic, H is cyclic
- If G has a subgroup of order n, then H has a subgroup of order n
- the order of a in G = the order of Φ(a) in H

G, H groups with identity elements eG, eH respectively. if f: G→H is a homo, then....(4 things)

- f(eG) = eH
- f(a
^{-1}) = f(a)^{-1} - Im f is a subgroup of H
- if f is injective, then G is isomorphic to Im f

Cayley's Theorem

Every group G is isomorphic to a group of permutations.

Every finite group G of order n is...

isomorphic to a subgroup of the symmetric group S_{n}

Cycle of length K

A permutation θ in S is a cycle of length K if there exist elements a1, a2,..... aK in {1, 2,.... n} such that

θ (a1) = a2

θ (a2) = a3

θ (aK) = aK

Disjoint

two cycles are disjoint if they have no elements in common.

if θ and λ are disjoint cycles in Sn, then...

θ λ= λθ

Every permutation in Sn is...

the product of disjoint cycles.

The **order** of the permutation θ in Sn is....

the LCM of the lengths of the disjoint cycles.

Transposition

a cycle of length 2

Every permutation in Sn is the product of...

transpositions!

even permutation

a permutation is even if it can be expressed as an even number of transpositions.

odd permutation

a permutation is odd if it can be expressed as an odd number of transpositions.

Alternating group An

The group of al even permutations in Sn.

lAnl =

n!/2

congruence with K<G, a, b e G

a is congruent to b mod K if ab^{-1} is in K.

congruence class of a mod K

Ka (the right coset).

index

the index of H in G [G:H] is the number of right cosets of H in G.

Lagranges Theorem

G = finite group. K < G. Then, order(K) divides order(G), and order(G) = [G:K] * order(K).

p = prime, p > 0. If Order(G) = p, then...

G is isomorphic to Zp.

If Order (G) = 4, then....

G is isomorphic to Z4 or V.

Normal

N<G is normal in G if Na = aN for every a in G.

G/N

Quotient Group: The set of all right cosets of N in G, if N is normal in G.

Kernel

Let f: G to H be a gp homo. Then, the kernel of f is: {a in g such that f(a) = e_{H}}

First Isomorphism Theorem

if f: G → H is a group homo, then G/ ker f ≅H.

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