Questions and Answers
What is a group?
A group 'G' or (G,*) is a set G with a binary operation * that satisfies associativity, has an identity element e, and every element has an inverse.
What is a binary operation?
'' on S is a rule assigning each ordered pair (a,b) of elements in S an element ab that is also in the set S.
What does it mean for a binary operation to be commutative?
In (S,), * is commutative if ab = b*a for every a, b ∈ S.
What does it mean for a binary operation to be associative?
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How do we compute a binary operation given by a table?
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What is an Abelian group?
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How do we know if a binary operation is commutative?
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How do we know if a binary operation is associative?
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Is # a binary operation on N, defined by a#b=c, where c is at least 5 more than a+b?
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What is a matrix?
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What matrix is a group?
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How do we multiply matrices?
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What is the General Linear Group?
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Are all matrices groups?
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What is the cancellation law theorem for groups?
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What is the solution theorem for groups?
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How many identity elements does a group have?
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How many inverses does an element in a group have?
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What is the distribution for the inverse of a*b?
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What does a finite group's multiplication table look like?
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What is a subgroup?
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What is Zn?
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What is an induced operation?
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What is the Klein-4 Group?
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What is the order of a group?
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Study Notes
Group
- A group ( G ) or ( (G,*) ) consists of a set ( G ) and a binary operation ( * ).
- The operation must be associative, possess an identity element ( e ), and each element must have an inverse.
Binary Operation
- A binary operation ( * ) on a set ( S ) assigns each ordered pair ( (a,b) ) to another element ( a*b ) in ( S ).
Commutative
- An operation ( * ) is commutative in ( (S,) ) if it satisfies ( ab = b*a ) for all ( a, b \in S ).
Associative
- An operation ( * ) is associative in ( (S,) ) if ( (ab)c = a(b*c) ) for all ( a, b, c \in S ).
Binary Operation Computation
- For a set ( S = {S_1, S_2, \ldots, S_n} ), the operation ( * ) is determined from a table where ( S_i * S_j ) corresponds to the entry in the ( i )-th row and ( j )-th column.
Abelian Group
- An Abelian group is one where the operation ( * ) is commutative.
Commutativity Check
- A binary operation is commutative if the operation table is symmetrical along the diagonal.
Associativity Check
- Associativity must be checked for all combinations of elements; finding one counterexample proves non-associativity.
Binary Operation on Positive Integers
- For ( a, b \in \mathbb{N} ), the operation ( a#b ) defined as ( c ) where ( c ) is at least 5 more than ( a+b ) is a binary operation on ( \mathbb{N} ).
Matrix
- A matrix ( M_{mn}(\mathbb{R}) ) is a collection of matrices with ( m ) rows and ( n ) columns, where matrix addition follows ( A + B = [a_{ij}] + [b_{ij}] ).
Group of Matrices
- ( (M_{mn}(\mathbb{R}), +) ) forms a group with the zero matrix as the identity element.
Matrix Multiplication
- Matrix multiplication ( AB ) is defined by the formula ( c_{ij} = \sum_{k=1}^{n} a_{ik} \times b_{kj} ).
General Linear Group
- The General Linear Group ( GL_n(\mathbb{R}) ) comprises matrices ( A ) for which there exists a matrix ( B ) such that ( AB = BA = I ), with ( I ) being the identity matrix.
Not All Matrices are Groups
- Some matrices lack inverses and do not form a group, particularly in ( M_{n}(\mathbb{R}) ).
Cancellation Law Theorem
- In a group ( (G,) ), the cancellation laws hold: if ( ab = a*c ), then ( b = c ).
Solution Theorem
- In a group ( (G,) ), the equations ( aX = b ) and ( Y*c = d ) have unique solutions ( X = a' * b ) and ( Y = d * c' ).
Identity Elements
- A group ( (G,*) ) has exactly one identity element ( e ).
Uniqueness of Inverses
- In a group ( (G,*) ), an element ( a ) has a unique inverse ( a' ).
Inverse Distribution
- The inverse of a product follows the rule ( (a*b)^{-1} = b^{-1} * a^{-1} ).
Finite Group Multiplication Table
- In a finite group, the first two rows and the first two columns of the multiplication table are identical, and each element appears exactly once in each row and column.
Subgroup
- A subgroup ( H \subset G ) is formed if ( (H,*) ) itself satisfies the group properties.
Set of Remainders
- ( \mathbb{Z}_n ) is the set of remainders modulo ( n ), with operations defined on the equivalence classes.
Solving in ( \mathbb{Z}_3 )
- In ( \mathbb{Z}_3 ), computations yield remainders based on the modular relationships of integers.
Induced Operation
- An operation ( * ) in ( (G,*) ) is induced by a subset ( S ) if for any ( S_1, S_2 \in S ), ( S_1 * S_2 \in S ).
Klein-4 Group
- The Klein-4 group consists of four elements ( V = {e, a, b, c} ) with commutativity and diagonals filled with the identity.
Order of a Group
- The order of a group is the total number of its elements.
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