Any one of these of prime order is also cyclic.
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