This has several consequences.
All cyclic groups are abelian since a cyclic group is generated by a single element, somay be writtenwithThe group is abelian since ifthenAn abelian group may always be constructed in this way.
The group table is symmetric about the main diagonal since the element in row i, column j, writtenso the elements in positionsandare identical.
If the group table is symmetric then the group is abelian sincefor all
There are some results connected with these two results.
Any group with a prime number of elements is abelian. It is generated by a single element, the powers of which must commute.
Examples of Abelian Groups
All rotation groups in the plane are abelian.
The integers under addition or multiplication.
Addition or multiplication modulo n (if a group is present).
The real numbers under addition or multiplication.
Matrices under addition
Complex ij numbers under multiplication or addition
Hamiltonian ijk numbers under addition.
The Klein Group consisting of the group of symmetries of the rectangle.
Examples of Non Abelian Groups
Rotation groups in more than two dimensions.
The real numbers under addition or multiplication.
The group of invertible matrices under multiplication
The non zero Hamiltonian ijk numbers under multiplication.
The general dihedral groups consisting of the group of symmetries of regular polygons.
The general symmetric group Sn consisting of all permutations of the numbers
]]>Higher order terms which include factors called Bernoulli numbers.
Higher order terms again include factors called Bernoulli numbers.
forHigher order terms include factors called Euler numbers.
forHigher order terms include Bernoulli numbers as factors.
for
for
]]>Taylor series have many purposes, from approximating a function, allowing approximate solutions of equations to be found, in complex analysis, analysis, and many more areas. Taylor series can be multiplied or divided to find the Taylor series of products or quotients of functions, inverted to find the Taylor series of inverse functions, taken around many different points if the Taylor series about a point does not converge.
A Taylor series around the pointis also called a Maclaurin series.
Taylor series can be manipulated in some basic ways:
]]>Classifying a group means classifying the group up to isomorphism. To classify groups of order 4, we can start by looking at the possible orders of the elements. The only numbers which divide 4 are 4, 2, 1. If the group has an element a of order 4, it is cyclic and is isomorphic to
If there is no element of order 4 there must be three elements of order 2. There can only be one element of order 1 – the identity. Any element of order 2 is its own inverse so the group is abelian ( ifthensobutandso) andfor every
If three elements of the group arethe fourth element must beand this must be equal toNeither of these can be equal tosince ifthenbut sinceis self inverse,contradicting the fact that the inverse of an element is unique. The group is isomorphic to the group of symmetries of the rectangle,also called the Klein group.
There are no other groups of order 4.
]]>Ifhas an element of order 6, it is cyclic and is isomorphic to
Supposehas no element of order 6, but has an elementof order 3, thencontains the elementsThere must be some elementso by composingwithandwe obtain the setBy cancellation none of these elements can be equal so these elements constitute the group. We can partially fill out the group table as below.





















can't be equalby cancellation (ifthen), and can't beor (since each ealement can only appear once in each row and column. We are left withor
Supposethen
so the order ofis 6 and the group is cyclic, but the group hase no element of order 6. The same analysis is done for the caseso that
Now consider the productin the second row, the same row asorand the same column asso can't be equal to any of them, soor
SupposeThe powers ofare
since
and the order ofis 6. As before this is a contradiction sinceis not cyclic, so
The other possibility is
We then have
The group table is then


This group isomorphic to the permutation groupor the group of symmetries of the equilateral triangle, also called
]]>If the grouphas orderwe may write
This necessarily means that all elements of cyclic groups commute and and that cyclic groups abelian, since ifandfor someso that
This then means that the Cayley table has a line of symmetry about the leading diagonal, as shown below for the rotation group of a regular hexagon.
The abelian property is inherited by all subgroups ofas is the symmetry property of the Cayley table.
]]>Typically we have to express a complex fractionin the formWe do this by multiplying top and bottom by the complex conjugate of the denominator, remembering thatThe complex conjugate of
Example: Expressin the form
Argand Diagrams
We may also have to plot complex numbers on an Argand diagram. This is a normal set of axes:is plotted as the pointIn the diagram below the complex numbers plotted as the point
Magnitudes, Arguments and the Polar Form of Complex Numbers
The magnitude of
the argument of
The polar form ofis written as
Multiplying Complex Numbers
Given two complex numbersandwe can find the product
We can express this in polar form as above,
then
Or we can expressandin polar form then using the normal rules for multiplying exponentials:
so
Dividing Complex numbers
We can use the method of the top of the page to express in cartesian form, or, if we require polar form, using the normal rule for dividing exponentials:
]]>Every rotation group of orderconsisting of the rotations of the regular polygon withsides. The elementmay be taken to be the element that rotates the polygon byWe writemeaning that G is generated by the element
We may writeit terms of its distinct elements, all form by composition of a with itself.
Every elepment ofis found by repeated compositions ofwith itself.
An elemtentthat generates a cyclic group of ordermust have order sothe identity element, andfor any
Every subgroup of a cyclic group is also cyclic, and the elements of each cyclic subgroup of a cyclic group with generatorof orderis formed by repeated compositions ofwith itself, for someIf the subgroup generated byhas order s, thenso that
The Cayley table for the rotation symmetries of a regular hexagon is given below.
Notice that each row is displaced one to the left of the row above it and wrapped, so that the leftmost element reappears on the right.
]]>The theorem is easy to prove using the relationship Raising both sides of this expression to the power ofgives
The theorem is useful when deriving relationships between trigonometric functions. For example, we can obtain polynomial expressions for sin n %theta and cos n %theta for any n using de Moivre's theorem.
Example: Derive expressions forandusing de Moivre's theorem.
(1)
Expanding the left hand side using the binomial theorem gives
(2)
Equating real coefficients of (1) and (2) gives respectively
Useto give Simplifying this expression gives
Equating imaginary coefficients of (1) and (2) gives respectively
Useto give Simplifying this expression gives
(1)
Expanding the left hand side using the binomial theorem gives
The real and imaginary parts of this expression are
Equating real parts gives
Useto give
This simplifies to
Equating imaginary parts gives
Useto give
This simplifies to
]]>Findingbecomes the problem of finding
Write
In general
hence
]]>