Groups

Definitions

Binary operation

A binary operation is a process that involves two members of a set and produces a result. For example, the operation could be defined as .

  • A binary operation is closed over set if , .
  • A binary operation is commutative over set if , .
  • A binary operation is associative over set if , .
  • A binary operation has an identity element in set if there exists a such that .
  • A binary operation has an inverse of element , denoted , in set with identity if there is an such that . An element is self-inverse if , and thus if .
Cayley table

A Cayley table shows the results of performing an operation with the element in the row first, and the element in the column second. A Cayley table is symmetric about the leading diagonal if the operation is commutative.

Latin square

A Latin square is a square table where each element appears exactly once in each row and column. The Cayley table of any group is a Latin square, but not all Latin squares represent a group[1].

Groups

A set under a binary operation forms a group if it satisfies the group axioms:

  • is closed under ,
  • the operation is associative,
  • there exists an identity in ,
  • and each element in has an inverse in .
    An Abelian group is a group where the binary operation is also commutative.

The order of a group is the number of elements in the group.
The order of an element is the power an element must be raised to in order to get the identity element. The order of an element is such that , where is the identity. The order of is 1.

Cyclic groups

A cyclic group is where each element of the group can be expressed as , where is a generator of the group and . A cyclic group can have multiple generators; any element with order equal to the order of the group is a generator.

Cyclic groups, denoted (where is the order of the group), have the following properties:

  • Commutative binary operation (Abelian group)
  • There must be at least one generator of order , such that
  • If the order of a group is , where is prime, then must be cyclic; groups of prime order must be cyclic.

Subgroups

A subgroup of a group is a subset of that satisfies the group axioms.

  • The trivial subgroup is a subgroup with just the identity element.
  • A proper subgroup is a subgroup of that is not itself (same idea as a proper subset).
    Any subgroup must contain the identity element. The identity element and any self-inverse element forms a subgroup.

Lagrange's theorem[2] states that the order of a subgroup must be a factor of the order of the group . Therefore:

  • The order of each element of is a factor of the order of (because each element can generate a cyclic subgroup such that the order of is equal to the order of the element, which must therefore be a factor of the order of ).
  • If has a prime order, then has no proper subgroups[3].

Groups to know

The specification requires knowledge of all groups with order . This includes cyclic groups, (the Klein 4-group), and (the symmetric group of order 3). The cyclic groups can be thought of as the set of integers under the operation of addition modulo .

Order 1

There is one trivial Abelian cyclic group of order 1, :

Order 2

There is one Abelian cyclic group of order 2, :

Order 3

There is one Abelian cyclic group of order 3, :

Order 4

There are two groups of order 4: the Abelian cyclic group and the Abelian Klein 4-group .

Where in , each element is self-inverse. can also be thought of as the symmetries of a rectangle, where are a rotation, or reflections in either axis (thus the elements are self-inverse).

Order 5

There is one Abelian cyclic group of order 5, .

Order 6

There are two groups of order 6: the Abelian cyclic group and the symmetric group (or the dihedral group )[4].

The symmetric group can be thought of as the group of all possible ways to rearrange three objects. is the same as , which can be thought of as the symmetries of an equilateral triangle (rotations and reflections across the three medians).

Notes on :

  • It is the smallest non-Abelian group, so the Cayley table is not symmetric about the diagonal.
  • In the above Cayley table, is the identity transformation can be thought of as reflections across the medians of an equilateral triangle, and and can be thought of as and rotations.
  • Note how are all self-inverse (as expected for reflections) and how and are inverses of each other (as expected for and rotations).

Order 7

There is one Abelian cyclic group of order 7, :

  1. Footnote on Latin Squares
    The Cayley table of any group is a Latin square, but not all Latin squares represent a group. A Latin square represents a quasigroup, where there is a closed operation and an inverse element for all elements, but there is not necessarily associativity or an identity.↩︎

  2. Footnote on Lagrange's Theorem
    This is A-level content, but it's useful at AS so it's here anyways.↩︎

  3. Footnote on proper subgroups and the trivial subgroup
    A group of prime order will still have the trivial subgroup. Whether the trivial subgroup is a proper subgroup is... debateable (the textbook says no, but Integral says yes).↩︎

  4. Footnote on / /
    The Internet can't agree on whether this group is or , so maybe just call it (OCR use ).↩︎

Interactive Graph
Table Of Contents
Groups
Definitions
Binary operation
Cayley table
Latin square
Groups
Cyclic groups
Subgroups
Groups to know
Order 1
Order 2
Order 3
Order 4
Order 5
Order 6
Order 7