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#include <iostream>
#include <set>
using namespace std;
//=========================================================
template <typename T> class Group
{
public:
set<T> S;
T (*op)( T, T );
void opTable();
bool isClosed();
bool isAssociative();
bool hasInverse();
bool hasIdentity( T &identity );
bool isCommutative();
};
//----------------------------
template <typename T> void Group<T>::opTable()
{
cout << "\t |\t";
for ( auto b : S ) cout << " " << b << '\t';
cout << "\n";
cout << "---\t---\t";
for ( auto b : S ) cout << "---" << '\t';
cout << "\n";
for ( auto a : S )
{
cout << a << "\t |\t";
for ( auto b : S ) cout << " " << op( a, b ) << '\t';
cout << '\n';
}
}
//----------------------------
template <typename T> bool Group<T>::isClosed()
{
for ( auto b : S )
{
for ( auto a : S )
{
if ( S.find( op( a, b ) ) == S.end() ) return false;
}
}
return true;
}
//----------------------------
template <typename T> bool Group<T>::isAssociative()
{
for ( auto c : S )
{
for ( auto b : S )
{
for ( auto a : S )
{
if ( op( a, op( b, c ) ) != op( op( a, b ), c ) ) return false;
}
}
}
return true;
}
//----------------------------
template <typename T> bool Group<T>::hasInverse()
{
T id;
if ( !hasIdentity( id ) ) return false; // pointless if no identity element
for ( auto b : S )
{
bool found = false;
for ( auto a : S )
{
if ( op( a, b ) == id && op( b, a ) == id )
{
found = true;
break;
}
}
if ( !found ) return false;
}
return true;
}
//----------------------------
template <typename T> bool Group<T>::hasIdentity( T &id )
{
id = T{};
auto a = *S.begin(); // find an identity for first element
bool found = false;
for ( auto b : S )
{
if ( op( b, a ) == a )
{
found = true;
id = b;
break;
}
}
if ( !found ) return false;
for ( auto b : S ) // identity must be unique
{
if ( op( id, b ) != b || op( b, id ) != b ) return false;
}
return true;
}
//----------------------------
template <typename T> bool Group<T>::isCommutative()
{
for ( auto b : S )
{
for ( auto a : S )
{
if ( op( a, b ) != op( b, a ) ) return false;
}
}
return true;
}
//=========================================================
// Some binary ops
int addModulo5 ( int a, int b ) { return ( a + b ) % 5; }
int add ( int a, int b ) { return a + b ; }
int timesmodulo5( int a, int b ) { return ( a * b ) % 5; }
int times ( int a, int b ) { return a * b ; }
//=========================================================
int main()
{
set<int> S = { 0, 1, 2, 3, 4 };
set<int> S0 = { 1, 2, 3, 4 };
// Group<int> G = { S, addModulo5 }; // *** abelian group ***
// Group<int> G = { S, add }; // *** not a group ***
// Group<int> G = { S , timesmodulo5 }; // *** not a group ***
Group<int> G = { S0, timesmodulo5 }; // *** abelian group ***
// Group<int> G = { S0, times }; // *** not a group ***
int id;
G.opTable();
bool Closed = G.isClosed();
bool Associative = G.isAssociative();
bool Inverse = G.hasInverse();
bool Neutral = G.hasIdentity( id );
bool Commutative = G.isCommutative();
cout << "G is " << ( Closed ? "" : "not " ) << "closed\n";
cout << "G is " << ( Associative ? "" : "not " ) << "associative\n";
cout << "G is " << ( Inverse ? "" : "not " ) << "invertible\n";
if ( Neutral )
{
cout << "G has an identity element " << id << '\n';
}
else
{
cout << "G has no identity element\n";
}
cout << "G is " << ( Commutative ? "" : "not " ) << "commutative\n";
if ( Closed && Associative && Inverse && Neutral )
{
cout << "G is " << ( Commutative ? "an abelian " : "a non-abelian " ) << "group\n";
}
else
{
cout << "G is not a group\n";
}
}
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