Ch. 4: Isomorphism

(2) These two squares do NOT have exactly the same symmetry groups:

(2) These two squares do NOT have exactly the same symmetry groups:

(Symmetries of the red square are NOT also symmetries of the green square

because they move the green square around.)

(2) These two squares do NOT have exactly the same symmetry groups:

BUT the squares are rigidly equivalent, so their symmetry groups

are essentially the same.

What should this mean? How are their Cayley tables similar?

Rene’s red square:

{I, R₉₀, R₁₈₀, R₂₇₀, H, V, D, D’}.

(Same Cayley table we built in Ch. 2)

Rene’s red square:

{I, R₉₀, R₁₈₀, R₂₇₀, H, V, D, D’}.

How can Gretchen copy Rene’s work to build her Cayley table?

Gretchen’s green square:

{I, R₉₀, R₁₈₀, R₂₇₀, F₁, F₂, F₃, F₄}.

Rene’s red square:

{I, R₉₀, R₁₈₀, R₂₇₀, H, V, D, D’}.

Gretchen’s green square:

{I, R₉₀, R₁₈₀, R₂₇₀, F₁, F₂, F₃, F₄}.

She should use this dictionary to convert as she copies!

Rene’s red square:

{I, R₉₀, R₁₈₀, R₂₇₀, H, V, D, D’}.

Gretchen’s green square:

{I, R₉₀, R₁₈₀, R₂₇₀, F₁, F₂, F₃, F₄}.

This creates a valid Cayley table for Gretchen (each cell is filled in correctly)!

This dictionary converts every true red equation into a true green equation!

H*D = R₉₀

F₁*F₄ = R₉₀

convert

DEFINITION: An isomorphism between two groups means a one-to-one matching (dictionary)

between their members that converts each true equation in one group into a true equation

in the other.

​

We say two groups are isomorphic there exists an isomorphism between them.

Thus, the symmetry group of the red square is isomorphic to

the symmetry group of the green square.

DEFINITION: An isomorphism between two groups means a one-to-one matching (dictionary)

between their members that converts each true equation in one group into a true equation

in the other.

​

We say two groups are isomorphic there exists an isomorphism between them.

EXERCISE: Prove that this dictionary is NOT an isomorphism between the red and green groups.

DEFINITION: An isomorphism between two groups means a one-to-one matching (dictionary)

between their members that converts each true equation in one group into a true equation

in the other.

​

We say two groups are isomorphic there exists an isomorphism between them.

EXERCISE: Prove that this dictionary is NOT an isomorphism between the red and green groups.

When a pair of finite groups are isomorphic, they have the same patterns in their Cayley tables. An observer who doesn’t know or care what the names of the group members represent would study the two Cayley tables and discover exactly the same patterns. From this observer’s perspective, the two groups would look like a single group represented in two different notational systems.

When a pair of finite groups are isomorphic, they have the same patterns in their Cayley tables. An observer who doesn’t know or care what the names of the group members represent would study the two Cayley tables and discover exactly the same patterns. From this observer’s perspective, the two groups would look like a single group represented in two different notational systems.

When a pair of finite groups are isomorphic, they have the same patterns in their Cayley tables. An observer who doesn’t know or care what the names of the group members represent would study the two Cayley tables and discover exactly the same patterns. From this observer’s perspective, the two groups would look like a single group represented in two different notational systems.

In fact, here is a description that characterizes both groups simultaneously: in addition to the identity, there are two more members. Each is the inverse of the other. Each combines with itself to give the other.

In fact, here is a description that characterizes both groups simultaneously: in addition to the identity, there are two more members. Each is the inverse of the other. Each combines with itself to give the other.

To what familiar group are these two groups isomorphic?

QUESTION: Do the star and the moth have isomorphic symmetry groups?

QUESTION: Do the star and the moth have isomorphic symmetry groups?

ANSWER: Yes! Here are their Cayley tables:

And here is the isomorphism:

QUESTION: Do the star and the moth have isomorphic symmetry groups?

ANSWER: Yes! Here are their Cayley tables:

And here is the isomorphism:

Q: Are these two groups isomorphic:

​

D₄ = The symmetry group of a square.

​

C₅ = The symmetry group of an oriented 5 sided polygon (or star).

Q: Are these two groups isomorphic:

​

D₄ = The symmetry group of a square.

​

C₅ = The symmetry group of an oriented 5 sided polygon (or star).

NO! They have different sizes.

Q: Are these two groups isomorphic:

​

D₄ = The symmetry group of a square.

​

C₈ = The symmetry group of an oriented 8-sided polygon (or star).

NO! Because only C₈ is commutative.

Q: Are these two groups isomorphic:

​

D₄ = The symmetry group of a square.

​

C₈ = The symmetry group of an oriented 8-sided polygon (or star).

Q: Are these two groups isomorphic:

​

D₂ = The symmetry group of a rectangle.

​

C₄ = The symmetry group of an oriented 4-sided polygon (or star).

Q: Are these two groups isomorphic:

​

D₂ = The symmetry group of a rectangle.

​

C₄ = The symmetry group of an oriented 4-sided polygon (or star).

No. Think about why no matching would work.

PROOF:

• A commutative group could never be isomorphic to a non-commutative group.

PROOF:

• A commutative group could never be isomorphic to a non-commutative group.

PROOF:

• A commutative group could never be isomorphic to a non-commutative group.
• Isomorphic groups always have the same size.

PROOF:

• A commutative group could never be isomorphic to a non-commutative group.
• Isomorphic groups always have the same size.

You have a large collection of bounded objects with finite symmetry

groups. Should you sort them into piles according to…

(1) rigid equivalence, or

(2) whether their symmetry groups are isomorphic.

You have a large collection of bounded objects with finite symmetry

groups. Should you sort them into piles according to…

(1) rigid equivalence, or

(2) whether their symmetry groups are isomorphic.

​

The only difference is whether these two piles merge into one pile.

You have a large collection of bounded objects with finite symmetry

groups. Should you sort them into piles according to…

(1) rigid equivalence, or

(2) whether their symmetry groups are isomorphic.

​

The only difference is whether these two piles merge into one pile.

Sorting is NOT the main purpose for isomorphisms in this book.

Q: Do these two border patterns have isomorphic symmetry groups?

G G G G G G G G G G G G G G G G G G G G G

P P P P P P P P P P P P P P

1 cm between Gs

2 cm between Ps

Q: Do these two border patterns have isomorphic symmetry groups?

G G G G G G G G G G G G G G G G G G G G G

P P P P P P P P P P P P P P

1 cm between Gs

2 cm between Ps

YES! Here is an isomorphism:

Symmetries of G-pattern

Symmetries of P-pattern

“T” means translate the subscripted number of centimeters (right if positive, left if negative).

Q: Do these two border patterns have isomorphic symmetry groups?

G G G G G G G G G G G G G G G G G G G G G

P P P P P P P P P P P P P P

1 cm between Gs

2 cm between Ps

YES! Here is an isomorphism:

Symmetries of G-pattern

Symmetries of P-pattern

T₅ * T₈ = T₁₃

↓ ↓ ↓

T₁₀ * T₁₆ = T₂₆

Watch this dictionary turn a true green

equation into a true purple equation:

G G G G G G G G G G G G G G G G G G G G G

Q: Are these two groups isomorphic:

​

• the symmetry group of the G-border pattern above

​

• Z = {…, -3, -2, -1, 0, 1, 2, 3,…} “the additive group of all integers”

G G G G G G G G G G G G G G G G G G G G G

YES! Here is an isomorphism:

Symmetries of G-pattern

The integers

Q: Are these two groups isomorphic:

​

• the symmetry group of the G-border pattern above

​

• Z = {…, -3, -2, -1, 0, 1, 2, 3,…} “the additive group of all integers”

G G G G G G G G G G G G G G G G G G G G G

YES! Here is an isomorphism:

Symmetries of G-pattern

The integers

Watch this dictionary turn a true green

equation into a true purple equation:

T₅ * T₈ = T₁₃

↓ ↓ ↓

5 + 8 = 13

Q: Are these two groups isomorphic:

​

• the symmetry group of the G-border pattern above

​

• Z = {…, -3, -2, -1, 0, 1, 2, 3,…} “the additive group of all integers”

EXERCISE: Prove that the symmetry group of this border pattern is

isomorphic to Z = {…, -3, -2, -1, 0, 1, 2, 3,…}

“the additive group of all integers”

EXERCISE: Prove that the symmetry group of this border pattern is

isomorphic to Z = {…, -3, -2, -1, 0, 1, 2, 3,…}

“the additive group of all integers”

Symmetries of pattern

Here is one system for naming all of the symmetries, including

the translations and the glide-reflections (where H = horizontal reflection).

One letter

EXERCISE: Prove that the symmetry group of this border pattern is

isomorphic to Z = {…, -3, -2, -1, 0, 1, 2, 3,…}

“the additive group of all integers”

Symmetries of pattern

Here is one system for naming all of the symmetries, including

the translations and the glide-reflections (where H = horizontal reflection).

Composition examples … why is it just like integer addition?

One letter

EXERCISE: Prove that the symmetry group of this border pattern is

isomorphic to Z = {…, -3, -2, -1, 0, 1, 2, 3,…}

“the additive group of all integers”

Symmetries of pattern

Here is one system for naming all of the symmetries, including

the translations and the glide-reflections (where H = horizontal reflection).

Composition examples … why is it just like integer addition?

One letter

EXERCISE: Prove that the symmetry group of this border pattern is

isomorphic to Z = {…, -3, -2, -1, 0, 1, 2, 3,…}

“the additive group of all integers”

Symmetries of pattern

The integers

Here is an isomorphism.

EXERCISE: Prove that the symmetry group of this border pattern is

isomorphic to Z = {…, -3, -2, -1, 0, 1, 2, 3,…}

“the additive group of all integers”

Symmetries of pattern

The integers

Here is an isomorphism. But a simpler naming system makes it

easier to verify it’s an isomorphism, since you can distinguish

reflections vs. translations just based on even vs. odd…

EXERCISE: Prove that the symmetry group of this border pattern is

isomorphic to Z = {…, -3, -2, -1, 0, 1, 2, 3,…}

“the additive group of all integers”

Symmetries of pattern

The integers

Here is an isomorphism. But a simpler naming system makes it

easier to verify it’s an isomorphism, since you can distinguish

reflections vs. translations just based on even vs. odd…

​

Here S_n means the only symmetry that shifts n letters.

This “slide-and-tilt” is a rigid motion of the plane.

​

We used it to create our isomorphism between their symmetry groups.

This “slide-and-tilt” is a rigid motion of the plane.

​

We used it to create our isomorphism between their symmetry groups.

THEOREM: If two objects are rigidly equivalent, then their symmetry groups are isomorphic.

This theorem is the real reason that the red and green

squares have isomorphic symmetry groups.

THEOREM: If two objects are rigidly equivalent, then their symmetry groups are isomorphic.

Q: Are these two stars rigidly

equivalent?

THEOREM: If two objects are rigidly equivalent, then their symmetry groups are isomorphic.

Q: Are these two stars rigidly

equivalent?

YES!

(so their symmetry groups

must be isomorphic)

THEOREM: If two objects are rigidly equivalent, then their symmetry groups are isomorphic.

Q: Are the star and the moth rigidly

equivalent?

THEOREM: If two objects are rigidly equivalent, then their symmetry groups are isomorphic.

Q: Are the star and the moth rigidly

equivalent?

NO!

(but their symmetry groups are isomorphic anyways)

THEOREM: If two objects are rigidly equivalent, then their symmetry groups are isomorphic.

Q: Are these two border patterns rigidly equivalent?

G G G G G G G G G G G G G G G G G G G G G

P P P P P P P P P P P P P P

THEOREM: If two objects are rigidly equivalent, then their symmetry groups are isomorphic.

Q: Are these two border patterns rigidly equivalent?

G G G G G G G G G G G G G G G G G G G G G

P P P P P P P P P P P P P P

NO!

But their symmetry groups are isomorphic anyways.

(notice that they have the same symmetry type.)

Two notation systems for

C₅ = this star’s five rotation symmetries

1

2

C₅ = {I, R₇₂, R₁₄₄, R₂₁₆, R₂₈₈}

C₅ = {0, 1, 2, 3, 4}

How many “turns”

How many degrees

Two notation systems for

C₅ = this star’s five rotation symmetries

1

2

C₅ = {I, R₇₂, R₁₄₄, R₂₁₆, R₂₈₈}

C₅ = {0, 1, 2, 3, 4}

How many degrees

How many “turns”

Which system do you prefer?

Two notation systems for

C₅ = this star’s five rotation symmetries

1

2

C₅ = {I, R₇₂, R₁₄₄, R₂₁₆, R₂₈₈}

C₅ = {0, 1, 2, 3, 4}

How many degrees

How many “turns”

R₂₁₆* R₂₈₈ = R₁₄₄

3 + 4 = 2 (in C₅)

Two notation systems for

C₅ = this star’s five rotation symmetries

1

2

C₅ = {I, R₇₂, R₁₄₄, R₂₁₆, R₂₈₈}

C₅ = {0, 1, 2, 3, 4}

How many degrees

How many “turns”

R₂₁₆* R₂₈₈ = R₁₄₄

3 + 4 = 2 (in C₅)

Convention: We’ll use this simpler notation system from now on.

​

The members of C_n will be denoted:

​

C_n = {0, 1, 2, 3, …, n-1}.

​

We’ll write “+” for the algebraic operation in C_n.

​

We’ll think of this as “addition with wrap-around”.

This new system is much better for

C₇ = the symmetry group of an oriented

seven-pointed star.

R_2(360/7) * R_3(360/7) = R_5(360/7)

2 + 3 = 5 (in C₇)

OLD

NEW