Group theory

• A binary operation combines two elements in a set and must be defined for all elements in the set
• An operation is commutative if the order that the elements are combined has no effect on the output
• An operation is associative if, in a series of elements all being combined in the same way, the pairs can be combined in any order without changing the result
• The identity element is an element which leaves other elements unchanged when it is combined with them using the given operation
• The inverse of an element is the value that you can combine with the element to get the identity
• An operation is closed on a set if all combinations of elements in the set result in a value in the set
• A set forms a group if:

  • The operation is closed
  • The operation is associative
  • The operation has an identity
  • Every element in the set has an inverse
• A group can be considered an Abelian group if the operation is also commutative
• A group in a Cayley table is a Latin Square - each element occurs exactly once in each row and column
• A subgroup is a subset of elements of a group that still forms a group under the same operation.
• The order of an element is the number of times that an element must be combined with itself to get the identity.
• The order of a group is the number of elements in the group (including the identity).
• A group that can be created by combining one element with itself multiple times is cyclic, and this element is a generator.
• Lagrange’s theorem states that, for a finite group, all subgroups have an order which is a factor of the order of the group. This also applies to orders of the elements of a group.
• The following are examples of groups, as well as the integers modulo n:

  • Order 4 - symmetries of a rectangle
  • Order 6 - symmetries of an equilateral triangle