Abstract Algebra - Groups Flashcards
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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?

<p>In (S,<em>), * is associative if (a</em>b)<em>c = a</em>(b*c) for every a, b, c ∈ S.</p> Signup and view all the answers

How do we compute a binary operation given by a table?

<p>If S = {S₁, S₂, ..., Sn}, we define * by the table where Si * Sj is the entry located at the ith row and jth column.</p> Signup and view all the answers

What is an Abelian group?

<p>A group where * is commutative.</p> Signup and view all the answers

How do we know if a binary operation is commutative?

<p>The table will be symmetrical along the diagonal.</p> Signup and view all the answers

How do we know if a binary operation is associative?

<p>We must check every combination of elements.</p> Signup and view all the answers

Is # a binary operation on N, defined by a#b=c, where c is at least 5 more than a+b?

<p>Yes.</p> Signup and view all the answers

What is a matrix?

<p>The matrix Mmn (R) is the set of all matrices with m rows and n columns.</p> Signup and view all the answers

What matrix is a group?

<p>(Mmn(R), +) is a group, where e is the zero matrix.</p> Signup and view all the answers

How do we multiply matrices?

<p>AB = [cij] where cij = ∑(k=1~n) aik × bkj.</p> Signup and view all the answers

What is the General Linear Group?

<p>GLn(R) = {A∈Mn(R) | ∃B∈Mn(R) such that AB = BA = I}.</p> Signup and view all the answers

Are all matrices groups?

<p>False (B)</p> Signup and view all the answers

What is the cancellation law theorem for groups?

<p>If a<em>b = a</em>c then b = c.</p> Signup and view all the answers

What is the solution theorem for groups?

<p>In a group (G,<em>), the equations a</em>X = b and Y*c = d have unique solutions X = a'<em>b and Y = d</em>c'.</p> Signup and view all the answers

How many identity elements does a group have?

<p>There is only one identity element 'e'.</p> Signup and view all the answers

How many inverses does an element in a group have?

<p>The inverse of any element a is unique.</p> Signup and view all the answers

What is the distribution for the inverse of a*b?

<p>(a*b)⁻¹ = b⁻¹ * a⁻¹.</p> Signup and view all the answers

What does a finite group's multiplication table look like?

<p>The uppermost two rows and leftmost two rows are identical.</p> Signup and view all the answers

What is a subgroup?

<p>H is a subgroup of G if (H,*) is itself a group.</p> Signup and view all the answers

What is Zn?

<p>Zn is the set of remainders when dividing by n.</p> Signup and view all the answers

What is an induced operation?

<p>S is closed under * if S₁, S₂∈S → S₁*S₂∈S.</p> Signup and view all the answers

What is the Klein-4 Group?

<p>V={e,a,b,c}, forming a commutative table where all the diagonals are e.</p> Signup and view all the answers

What is the order of a group?

<p>The order of a group is the number of elements in the group.</p> Signup and view all the answers

Flashcards

Group

A set G with a binary operation * that is associative and has an identity and inverses.

Binary Operation

A binary operation * assigns each pair (a,b) to an element a*b in set S.

Commutative

An operation * is commutative if ab = ba for all a, b in S.

Associative

An operation * is associative if (ab)c = a(bc) for all a, b, c in S.

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Abelian Group

An Abelian group is a group where the operation * is commutative.

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Commutativity Check

To check commutativity, see if the operation table is symmetrical.

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Associativity Check

To check for associativity, find a counterexample through combinations.

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Identity Element

A unique element e in a group G such that a*e = a for all a in G.

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Inverse Element

An inverse a' in a group G satisfies a*a' = e, the identity.

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Unique Inverses

In a group G, each element has exactly one inverse a'.

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Inverse Distribution

The inverse of a product follows (a*b)^{-1} = b^{-1} * a^{-1}.

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Finite Group Table

In finite groups, the multiplication table has identical first two rows and columns.

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Subgroup

A subgroup H is a subset of G that itself meets group properties.

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Order of a Group

The order of a group is the total count of its elements.

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Matrix

A matrix M_{mn} consists of m rows and n columns, used in operations.

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Matrix Addition

Matrix addition follows element-wise by A + B = [a_{ij}] + [b_{ij}].

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Group of Matrices

The set (M_{mn}( ext{R}), +) forms a group under addition.

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Matrix Multiplication

Matrix multiplication is defined by c_{ij} = ∑(a_{ik} * b_{kj}).

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General Linear Group

GL_n(R) includes matrices A where AB = BA = I, the identity matrix.

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Not All Matrices are Groups

Some matrices lack inverses and do not form groups in M_n(R).

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Cancellation Law Theorem

If ab = ac, then b = c in a group.

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Solution Theorem

The equations aX = b and Yc = d have unique solutions in a group.

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Set of Remainders

ℤ_n is the set of remainders modulo n, with defined operations.

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Induced Operation

An operation * in (G,*) is induced if S_1 * S_2 remains in S.

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Klein-4 Group

The Klein-4 group has four elements {e, a, b, c} with commutativity.

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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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Test your understanding of key concepts in Abstract Algebra with these flashcards focusing on groups and binary operations. Each card includes definitions and essential properties that are crucial for mastering the subject. Ideal for students looking to reinforce their knowledge.

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