Unit - 1

Symmetries of a square

1.1 Symmetries of a square

In order to understand the idea of symmetry, we begin with a very simple example. The equilateral triangle below. What does it mean for this shape to be symmetric?

Rotation symmetries

An equilateral triangle can be rotated by 120 degree, 240 degree, or 360 degree angles without changing it.

If we rotate the triangle by 31 degree or 87 degree we notice that the bottom edge of the triangle is no longer perfectly horizontal.

Many other shapes that are not regular polygons also have rotational symmetries.

Several shapes with rotational symmetries are given below

The first shape can be rotated any integer multiple of 72 degree.

We say that a shape has rotational symmetry of order n if it can be rotated by any multiple of /n without changing its appearance. We can imagine constructing other shapes with rotational symmetries of arbitrary order. If the only rotations that leaves a shape unchanged are multiples of 360, then we say that the shape has only the trivial (order n = 1) symmetry.

Mirror reflection symmetries

Another type of symmetry that we can find in two-dimensional geometric shapes is mirror reflection symmetry. More specifically, we can draw a line through some shapes and reflect the shape through this line without changing its appearance. This is called a mirror reflection symmetry.

Rotational symmetries and mirror reflection symmetries are not exclusive, and the same shape can have symmetries of both kinds. The equilateral triangle clearly has both mirror reflection symmetries and rotational symmetries.

Symmetries of the square

A square is more symmetric than a triangle. Any two squares in the same row can be obtained from one another through rotations, whereas those in distinct rows can only be obtained from one another through a reflection.

If we remove a square region from a plane, move it in some way, then put the square back into the space it originally occupied.

Here we need to describe the possible relationships between the initial position of the square and its final position in terms of motions.

The final position of the square is completely determined by the location and orientation (that is, face up or face down) of any particular corner. But, clearly, there are only four locations and two orientations for a given corner, so there are exactly eight distinct final positions for the corner.

1. Rotation of 0° (no change in position)

2.     Rotation of 90° (counterclockwise)

3.     Rotation of 180°

4.     Rotation of 270°

5.     Flip about a horizontal axis

6.     Flip about vertical axis

7.     Flip about the main diagonal

8.     Flip about the other diagonal

Key takeaways:

1. An equilateral triangle can be rotated by 120 degree, 240 degree, or 360 degree angles without changing it.
2. Some shapes and reflect the shape through this line without changing its appearance. This is called a mirror reflection symmetry.
3. A square is more symmetric than a triangle. Any two squares in the same row can be obtained from one another through rotations, whereas those in distinct rows can only be obtained from one another through a reflection.

1.2 Dihedral groups

The dihedral group  is the symmetry group of an -sided regular polygon for . The group order of  is . Dihedral groups  are non-Abelian permutation groups for .

Note- The dihedral group of order 2n is often called the group of symmetries of a regular n-gon.

A symmetry group consisting of the rotational symmetries of , , , . . . , (n-1), and no other symmetries is called a cyclic rotation group of order n and is denoted by .

One group presentation for the dihedral group  is .

A reducible two-dimensional representation of  using real matrices has generators given by  and , where  is a rotation by  radians about an axis passing through the center of a regular -gon and one of its vertices and  is a rotation by  about the center of the -gon 

Dihedral groups all have the same multiplication table structure. The table for  is illustrated above.

The cycle index (in variables , ..., ) for the dihedral group  is given by

 (1)

Where

Is the cycle index for the cyclic group ,  means  divides , and  is the totient function. The cycle indices for the first few  are

   (3)

  (4)

  (5)

  (6)

Key takeaways

1. The dihedral group  is the symmetry group of an -sided regular polygon for . The group order of  is .
2. The dihedral group of order 2n is often called the group of symmetries of a regular n-gon.

1.3 Definition and examples of groups including permutation groups and quaternion groups (illustration through matrices)

Binary operation-

Let G be a set. A binary operation on G is a function that assigns each ordered pair of elements of G an element of G.

A binary operation on a set G, then, is a method by which the members of an ordered pair from G combine to yield a new member of G. This condition is known as closure. The most familiar binary operations are ordinary addition, subtraction, and multiplication of integers. Division of integers is not a binary operation on the integers because an integer divided by an integer need not be an integer.

Group- A group is an algebraic structure (G, *) in which the binary operation * on G satisfies the following conditions:

Condition-1:  for all a, b, c, ∈ G

a * (b * c) = (a * b) * c (associativity)

Condition-2: there exists an elements e ∈G such that for any a ∈G

a * e= e * a = a (existence of identity)

Condition-3: for every a ∈G, there exists an element denoted by in G such that

a * = * a = e

is called the inverse of a in G.

Example: (Z, +) is a group where Z denote the set of integers.

Example: The set {1, 2, . . . , n - 1} is a group under multiplication modulo n if and only if n is prime.

Example: (R, +) is a group where R denote the set of real numbers.

Example: The set Z of all integers forms a group with respect to addition; the identity element is 0 and the inverse of a ∈ Z is  -a.

Z is not a multiplicative group since, for example, neither 0 nor 2 has a multiplicative inverse.

Example: The set A = {1, -1, I, -i}  with respect to multiplication on the set of complex numbers forms a group.

Example: The set of integers under subtraction is not a group, since the operation is not associative.

Example: The set of integers under ordinary multiplication is not a group. Since the number 1 is the identity, property 3 fails. For example, there is no integer b such that 5b 5 1.

Example: The set S of positive irrational numbers together with 1 under multiplication satisfies the three properties given in the definition of a group but is not a group.

Example: The set Zn = {0, 1, . . . , n - 1} for n 1 is a group under

Addition modulo n. For any j . 0 in Zn, the inverse of j is n 2 j.

This group is usually referred to as the group of integers modulo n.

Abelian group-

Let (G, *) be a group. If * is commutative that is a* b = b * a for all a, b ∈ G then (G, *) is called an Abelian group.

Examples:

1. The set of all non-zero rational numbers with multiplication composition is an infinite Abelian group. 1 is the identity and 1/a is the inverse of a.
2. The set of all n-th roots of unity with multiplication composition is an finite abelian group of order n, where n is any given natural number.
3. The set of all unimodular complex numbers is an infinite abelian group with multiplication composition.

Finite group-

A group G is said to be a finite group if the set G is a finite set.

Infinite group-

A group G, which is not finite is called an infinite group.

Order of a finite group-

The order of a finite group (G, *) is the number of distinct element in G. The order of G is denoted by O (G) or by |G|.

Note- A group is called Abelian or non-abelian accordingly as the group operation is or is not commutative.

Example: If G = {1, -1, i, -i} where i= , then  show that G is an abelian group with respect to multiplication as a binary operation.

Sol.

First we will construct a composition table-

It is clear from the above table that algebraic structure (G, .) is closed and satisfies the following conditions.

Associativity- For any three elements a, b, c ∈G (a ⋅b) ⋅c = a ⋅(b ⋅c)

Since

1 ⋅(−1 ⋅i) = 1 ⋅−i= −i

(1 ⋅−1) ⋅i= −1 ⋅i= −i

⇒1 ⋅(−1 ⋅i) = (1 ⋅−1) i

Similarly with any other three elements of G the properties holds.

∴Associative law holds in (G, ⋅)

Existence of identity: 1 is the identity element (G, ⋅) such that 1 ⋅a = a = a ⋅1 ∀a ∈G

Existence of inverse: 1 ⋅1 = 1 = 1 ⋅1 ⇒1 is inverse of 1

(−1) ⋅(−1) = 1 = (−1) ⋅(−1) ⇒–1 is the inverse of (–1)

i⋅(−i) = 1 = −i⋅i⇒–iis the inverse of iin G.

−i⋅i= 1 = i⋅(−i) ⇒iis the inverse of –iin G.

Hence inverse of every element in G exists.

Thus all the axioms of a group are satisfied.

Commutativity: a ⋅b = b ⋅a ∀a, b ∈G hold in G

1 ⋅1 = 1 = 1 ⋅1, −1 ⋅1 = −1 = 1 ⋅−1

i⋅1 = i= 1 ⋅i; i⋅−i= −i⋅i= 1 = 1 etc.

Commutative law is satisfied

Hence (G, ⋅) is an abelian group.

Example-

Prove that the set Z of all integers with binary operation * defined by a * b = a + b + 1 ∀a, b ∈G is an abelian group.

Sol: Sum of two integers is again an integer; therefore a +b ∈Z ∀a, b ∈Z

⇒a +b + 1 ⋅∈Z ∀a, b ∈Z

⇒Z is called with respect to *

Associative law for all a, b, a, b ∈G we have (a * b) * c = a * (b * c) as

(a* b) * c = (a + b + 1) * c

= a + b + 1 + c + 1

= a + b + c + 2

Also

a* (b * c) = a * (b + c + 1)

= a + b + c + 1 + 1

= a + b + c + 2

Hence (a * b) * c = a * (b * c) ∈a, b ∈Z.

Permutation groups-

Suppose X is a non-empty set. The group of all permutations of X under composition of mapping is called the symmetric group on X and it is denoted by .

A subgroup of is called a permutation group on X.

We can see that a bijection induces in a natural way an isomorphism .

If |X| = n, is denoted by and it is called the symmetric group of degree n.

A permutation can be in the form

Having two rows of integers, on the top row there are integers- 1, . . . , n and the bottom row has below i for each i = 1,2..n.

Which is called the two-row notation for a permutation.

Group of matrices

1. The set of all m by n matrices of integer elements, is an infinite Abelian group with matrix addition as composition.
2. The set of all n b n non singular matrices having elements as rational numbers is an infinite non-abelian group with matrix multiplication as composition.

The set of n by n non singular matrices having elements as integers is not a group matrix multiplication, for not all such matrices are inversible,

For example:

Determinant of the above matrix is -4 is non-singular but non-invertible in as much there exists no 2 by 2 matrix with elements as integers whose product with the given matrix is the identity matrix.

Key takeaways

1. A binary operation on a set G, then, is a method by which the members of an ordered pair from G combine to yield a new member of G.
2. (Z, +) is a group where Z denote the set of integers.
3. The set of integers under subtraction is not a group, since the operation is not associative.
4. The set of all unimodular complex numbers is an infinite abelian group with multiplication composition.
5. A group G is said to be a finite group if the set G is a finite set.
6. A group is called Abelian or non-abelian accordingly as the group operation is or is not commutative.

1.5 Elementary properties of groups

General properties of groups

1. Suppose we have a group G, then

Where a ∈ G, b ∈ G and c ∈ G

We have

Similarly we can do for left cancellation law.

2.     For any given pair of elements a, b of a group G, there exists a unique elements x, y of G such that

3.     General associative law-

For a group, we have

We will now state and prove a generalisation of this associative law for any arbitrary set of n elements.

Suppose an ordered set

Of 4 elements, we can, without changing the order of the elements, consider a number of different associations giving to the corresponding product.

Thus we have

Here is the fact that as a consequences of the associative law (1) holding for three elements. All these products are equal.

The result holds for any arbitrary element.

4.     Inverse of the product of an arbitrary number of elements

Suppose a and b are any two elements of a group G, then we have

By using the general associative law,

And

By the principal of induction, and the general associative law, we show that

That means that the inverse of the product of any number of elements is equal to the product of the inverses taken in the reverse order.

5.     Indices laws

We can easily show that

For every a belongs to G and m, n are the integers.

Theorem: (Uniqueness of the Identity) - In a group G, there is only one identity element.

Proof:

Suppose both e and e9 are identities of G. Then,

1. a e = a for all a in G, and

2. e’a = a for all a in G.

The choices of a = e’ in (1) and a = e in (2) yield e’e = e’ and e’e = e. Thus, e and e’ are both equal to e’e and so are equal to each other.

Theorem: (Cancellation)- In a group G, the right and left cancellation laws hold; that is, ba = ca implies b = c, and ab = ac implies b = c.

Proof:

Suppose ba = ca. Let a’ be an inverse of a. Then, multiplying

On the right by a’ yields (ba)a’ = (ca)a’. Associativity yields b(aa’) = c(aa’). Then, be = ce and, therefore, b = c as desired. Similarly, we can prove that ab = ac implies b = c by multiplying by a’ on the left.

Key takeaways:

1. In a group G, there is only one identity element.
2. In a group G, the right and left cancellation laws hold; that is, ba = ca implies b = c, and ab = ac implies b = c.

1.6 Subgroups and examples of subgroups

Order of a group- The number of elements of a group (finite or infinite) is called its order. We will use |G| to denote the order of G.

Example: the group Z of integers under addition has infinite order, whereas the group U(10) 5 {1, 3, 7, 9} under multiplication modulo 10 has order 4.

Order of an Element- The order of an element g in a group G is the smallest positive integer n such that 5 e. (In additive notation, this would be ng = 0.) If no such integer exists, we say that g has infinite order. We denote the order of an element g is denoted by |g|.

Definition-

Let (G, *) be a group and H, be a non-empty subset of G. If (H, *) is itself is a group, then (H, *) is called sub-group of (G, *).

We use the notation H G to mean that H is a subgroup of G. If we want to indicate that H is a subgroup of G but is not equal to G itself, we write H G. Such a subgroup is called a proper subgroup. The subgroup {e} is called the trivial subgroup of G; a subgroup that is not {e} is called a nontrivial subgroup of G.

Example-Let a = {1, –1, i, –i} and H = {1, –1}

G and H are groups with respect to the binary operation, multiplication.

H is a subset of G, therefore (H, X) is a sub-group (G, X).

Theorem-

If (G, *) is a group and H ≤G, then (H, *) is a sub-group of (G, *) if and only if

(i) a, b ∈H ⇒a * b ∈H;

(ii) a∈H ⇒∈H

Proof:

If (H, *) is a sub-group of (G, *), then both the conditions are obviously satisfied.

We, therefore prove now that if conditions (i) and (ii) are satisfied then (H, *) is a sub-group of (G, *).

To prove that (H, *) is a sub-group of (G, *) all that we are required to prove is : * is associative in

H and identity e ∈H.

That * is associative in H follows from the fact that * is associative in G.

Also,

a∈H ⇒∈H by (ii) and e ∈H and ∈H ⇒a * = e ∈H by (i)

Hence, H is a sub-group of G.

Tests for the determination of subgroups

Theorem- one-Step Subgroup Test

Let G be a group and H a nonempty subset of G. If a is in H whenever a and b are in H, then H is a subgroup of G. (In additive notation, if a - b is in H whenever a and b are in H, then H is a subgroup of G.)

Proof:

Since the operation of H is the same as that of G, it is clear that this operation is associative. Next, we show that e is in H. Since H is nonempty, we may pick some x in H. Then, letting a = x and b = x in the hypothesis, we have e = x  5 a  is in H. To verify that  is in H whenever x is in H, all we need to do is to choose a 5 e and b 5 x in the statement of the theorem. Finally, the proof will be complete when we show that H is closed; that is, if x, y belong to H, we must show that xy is in H also. We have already shown that  is in H whenever y is; so, letting a = x and b = , we have xy = x  = a  is in H.

Example: Let G be an Abelian group with identity e. Then H = {x [ G | = e} is a subgroup of G. Here, the defining property of H is the condition = e. So, we first note that = e, so that H is nonempty.

Now we assume that a and b belong to H. This means that = e and = e. Finally, we must show that = e. Since G is Abelian, = =   = = e.

Therefore, belongs to H and, by the One-Step Subgroup Test, H is a subgroup of G.

Theorem- Two-Step Subgroup Test

Let G be a group and let H be a nonempty subset of G. If ab is in H whenever a and b are in H (H is closed under the operation), and is in H whenever a is in H (H is closed under taking inverses), then H is a subgroup of G.

Proof:

By above theorem, it suffices to show that a, b H implies H. So, we suppose that a, b H. Since H is closed under taking inverses, we also have H. Thus, H by closure under multiplication.

When applying the “Two-Step Subgroup Test,” we proceed exactly as in the case of the “One-Step Subgroup Test,” except we use the assumption that a and b have property P to prove that ab has property P and that has property P.

Example: Let G be the group of nonzero real numbers under multiplication, H = {x G | x = 1 or x is irrational} and K = {x G | x 1}. Then H is not a subgroup of G, since H but Also, K is not a subgroup, since  but "

Theorem- Finite Subgroup Test- Let H be a nonempty finite subset of a group G. If H is closed under the operation of G, then H is a subgroup of G.

Proof:

By the above theorem, we need only prove that whenever a [ H. If a 5 e, then = a and we are done. If a e, consider the sequence a, , . . . . By closure, all of these elements belong to H. Since H is finite, not all of these elements are distinct. Say = and i > j. Then, = e; and since a e, i - j > 1. Thus, = = e and, therefore, = . But, i - j - 1 1 implies H.

Key takeaways:

1. The number of elements of a group (finite or infinite) is called its order. We will use |G| to denote the order of G.
2. The order of an element g in a group G is the smallest positive integer n.
3. Let (G, *) be a group and H, be a non-empty subset of G. If (H, *) is itself is a group, then (H, *) is called sub-group of (G, *).

1.7 Centralizer, normalizer and center of a group.

Definition Center of a Group

The center, Z(G), of a group G is the subset of elements in G that commute with every element of G. In symbols,

Definition Centralizer of a in G

Let a be a fixed element of a group G. The centralizer of a in G, C(a), is the set of all elements in G that commute with a. In symbols, C(a) = {g  G | ga = ag}.

Note- For each a in a group G, the centralizer of a is a subgroup of G.

Definition normalizer of a in G

Let H be a subgroup of a group G. Then the normalizer of H in G is N(H)  = {x  G | x}.

Theorem: For any subgroup S of a finite group G, N(S) will be the largest subgroup of G that contains S as a normal subgroup.

Proof:

Now u S = S, so u N(S) and, hence, N(S) is not equals to . If a, b N(S), then  = a (   = = S. Thus, ab N(S) and N(S) will be a subgroup of G. So, by definition of N(S), S is a normal subgroup of N(S) and N(S) contains any subgroup that has S as a normal subgroup.

Key takeaways

1. The center, Z(G), of a group G is the subset of elements in G that commute with every element of G.
2. For each a in a group G, the centralizer of a is a subgroup of G.

References:

1. Contemporary abstract algebra, Joseph A. Gallian

2. Schaum’s abstract algebra

3. Basic abstract algebra by P.B. Bhattacharay, SK jain, S.R. Nagpaul

4. Modern abstract algebra by Shanti narayan, S.Chand & Co.

5. Https://www2.math.upenn.edu/~mlazar/math170/notes07.pdf

6. Https://mathworld.wolfram.com/DihedralGroup.html