Introduction to group theory
Goutam sanyal
1
Mathematical Structures
• Mathematical structure (system)
Such a collection of objects with operations defined on them and
the accompanying properties form a mathematical structure or
system, for instance,
Example 1: The collection of sets with the operations of union,
intersection and complement and their accompanying properties
is a mathematical structure. Denoted by
(sets, U, ∩ , -)
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Mathematical Structures
• Binary operation
An operation that combines two objects
• Unary operation
An operation that requires only one object
Example: the structure (5x5 matrices, +, *, T)
the operations + and * are binary operations
the operation T is a unary operation
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Mathematical Structures
• Closure
A structure is closed with respect to an operation if that
operation always produces another/same member of the
collection of objects.
Example 3: The structure (5x5 matrices, +, *, T) is closed with
respect to +, * and T. (why?)
Example 4: The structure (odd integers, +, *) is closed with
respected to *, while it is not closed with respected to +. (why?)
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. Mathematical Structures
• Commutative property
If the order of the objects does not affect the outcome of a binary
operation, we say that the operation is commutative , namely
if x □ y = y □ x, where □ is some binary operation with
commutative property.
Example 6
(a) Join and meet for Boolean matrices are commutative operations
A V B =B V A and A ^ B = B ^ A
(b) Ordinary matrix multiplication is not a commutative operation.
AB ≠ BA
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Mathematical Structures
• Associative property
if □ is a binary operation, then □ is associative or has
associative property if
(x □ y) □ z = x □ (y □ z)
Example 7
Set union is an associative operation, since
(A U B) U C = A U (B U C) is always true
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Mathematical Structures
• Distributive property
If a mathematical structure has tow binary operations, say □ and
∇, a distributive property has the following pattern:
x □ (y ∇ z) = (x □ y) ∇ ( x □ z )
we say that □ distributes over ∇
Example 8 (b)
the structure (sets, U, ∩, -) has two distributive properties:
A U (B ∩ C) =(A U B) ∩ (A U C)
A ∩ (B U C) =(A ∩ B) U (A ∩ C)
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Mathematical Structures
• Identity
If a structure with a binary operation □ contain an element e,
satisfying that
x□e=e□x=x
for all x in the collection
we call e an identify for the operation □
Example 10:
For (n-by-n matrices, +,*, T), In is the identity for matrix
multiplication and the n-by-n zero matrix is the identity matrix
addition.
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Mathematical Structures
• Inverse
If a binary operation □ has an identity e, we say y is a □-
inverse of x if x □y=y □x=e
Example 11:
(a) In the structure (3-by-3 matrices, +, *, T), each matrix A=[aij]
has +-inverse(additive inverse), -A=[-aij]. (why ?)
(b) In the structure (integers, +, *), only the integers 1 and -1
have multiplicative inverses. (why?)
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Group
• A nonempty set G with an operation * is called a
group if the following axioms are satisfied
– Closure Law : if a,b ∈G then a*b ∈G
– Associativity : if a,b,c ∈G then a*(b*c)=(a*b)*c
– Existence of identity element : There exit an element
e such that e*a=a*e=a
– Existence of Inverse: for each element a ∈G there
exit a-1 ∈G such that a* a-1 = a-1 *a=e
Integer under mod 5 addition is a group
Symmetry
Symmetry
Symmetry Group
Symmetry Group
THANK YOU
