Groups Flashcards
What are the group axioms?
Closure. (the operation is closed on the set).
Existence of an identity. (the identity for the operation is in the set).
Existence of inverses. (there exists an inverse for each element, both are in the set).
Associativity. (the operation is associative for all elements in the set).
What do the group axioms result in being the case for an group displayed on a cayley table?
Each element appears once in every row and once in every column. (latin square method)
What is an abelian group?
A group for which the operation is also commutative.
What is the order of a group?
The number of elements in the set.
What is the period of an element in a group?
The smallest number of repeated applications of the element to get the identity element.
e.g: the identity always has period 1.
What is a proper, non trivial subgroup?
Subgroup = subset of a group that also follows the group axioms. Proper = the subgroup has order less than the group. Non-trivial = the subgroup does not contain only the identity (order greater than 1).
What is a cyclic group?
If a cyclic group has order n, then at least one element has period n and the rest have period n or a factor of n.
What are the two key examples of the cyclic group order n?
The group formed by the operation addition modulo n on the set of integers from 0 to n-1.
The subgroup of the symmetry group of a regular n-sided polygon consisting of rotations but no reflections.
What can be deduced about a non-trivial group if it has no proper non-trivial subgroup?
It is a cyclic group of prime order.
What is Lagrange’s theorem?
For any finite group of order n, all subgroups must have order that is a factor of n.
What is a generator element of a group?
An element for which repeated application of that element forms every element in the group.
If an element of a group order n has period n, then the element is a generator.
Show that for a cyclic group of prime order, each element generates the group, other than the identity.
As the group is cyclic and of order n, where n is prime, each element has period 1 or n. Only the identity element has period 1, so every other element has period n, so generates the group.
What is the name given to groups that have corresponding structure?
They are isomorphic
What is the symmetry group of a regular n-sided polygon?
The set of the orientations of the polygon formed by rotating or reflecting the polygon.
The symmetry group has order 2n (n rotation and n reflections).
