Symmetric types of this algebraic structure consist of permutations of a given number of elements.
One of these constructs is called perfect if it is equal to its own commutator. The purpose of the Hölder program was to classify a certain type of these. Those of odd order are solvable according to the Feit-Thompson theorem. Any one of these of prime order is also cyclic. Symmetric types of this algebraic structure consist of permutations of a given number of elements. Examples of these include one that has a 194 by 194 character table created by Fisher and Greiss. One of these of order six is the smallest one of these that is non-commutative. They must be associative and have an identity, inverses and a zero element, and Abelian ones are commutative. For 10 points, name these algebraic structures that consist of a set and a binary operation performed on that set.
One of these constructs is called perfect if it is equal to its own commutator.
 groups