1-to-1 | A map from {x}->{y} in which every x is sent to a different y, i.e. x not equal y implies f(x) not equal f(y). This does not imply that the map must completely cover {y}. (See "onto", "bijection".) | Abelian | A group G is called abelian if a*b=b*a for all a,b in G. (See "boring.") | Automorphism | An automorphism is an ismomorphic map of a group G onto itself. The set of isomorphisms of a given group G is also a group, called Aut(G). | Bijection |
A bijection is a map which is both 1-to-1 and onto. Such maps take each element of {x} to a unique {y}, and each {y} is the result of a unique {x}; therefore, bijections are invertible maps and sometimes written {x} <-> {y} . |
boring | A group G is called "boring" if it isn't particularly related to any interesting puzzles, physics, or profound cool group stuff. At least in Jim's opinion. (see "Abelian".) | Campanology |
The art and science of "change ringing", ringing N
church (or hand) bells in many of their possible
N permutations in a systematic way. Folks who did this understood quite a bit about permutations long before the mathematics of "group theory" existed; therefore, you could say it was the precursor of all this stuff. |
Conjugate | Two group elements a and b are conjugate to each other if there exists an element c such that a = c b c-1 . The set of all {x} conjugate to a given element y is called y's "conjugacy class." It turns out that these conjugacy classes partition each group into disjoint subsets; each group element belongs to a single conjugacy class. | Cayley's Theorem (All groups are permutations.) |
Every group G is is isomorphic to a subgroup of SG, the
group of permutations of the elements of G. If G is finite and has order n, then G is isomorphic to a subgroup of Sn. |
Countable | A set X is countable iff there exists a 1-to-1 map from the positive integers to X, {1,2,3,...} <-> {X}. | Commute | Two group elements a,b are said to commute if a*b=b*a. (See "Abelian.") | Commutator | The commutator [a,b] of two elements is a*b*a-1*b-1. | Commutator Subgroup | The set of all {x} such that x = [a,b] for some a,b in a group G is called that group's commutator subgroup. The order of this subgroup is a measure of how abelian-like G is. | Coset |
Given a group G with a subgroup H={h1,h2,...}, the "left coset"
of H corresponding to an element x of G is defined as the set
{ x h1 , x h2 , x h3, ... }. "x is in the same coset as y" defines an equivalence relation between x and y, and thus partitions G into order(H) disjoint sets. Showing that each of these cosets has the same number of elements leads to a proof of Lagrange's Theorem. |
cubelet | One of the smaller solid cubes which together make up the Rubik's Cube puzzle. Each of the corners, edges, and faces in the 3x3x3 Rubik's Cube is a "cubelet." | Cyclic | The cyclic groups ( Cn ) are those isomorphic to the integers {0,1,2,3,...,(n-1)} under addition mod n. | Determinant |
The determinant of an NxN square matrix is the scalar value of
the N-dimensional "volume" spanned by the column vectors of
the matrix. In particular, if the determinant is zero then
the matrix has no inverse, is is not particularly interesting. For a 2x2 matrix (a,b; c,d) the determinant is a*d-b*c. For larger matrices the formulas get trickier; check any linear algebra or calculus text. |
Examples |
Some specific named groups discussed in class and (brief) definitions:
|
Equivalence relation |
An equivalence relation on a set S is a set of pairs
of elements (s1,s2) with the following
properties:
|
Function | A function F(x)=y is a map between two sets, {x}->{y}. (See Map.) | Generators |
The generators of a group G are elements in a subset H of G = {h1,h2,h3...}
such that any element of G may be reached by a some sequence
x1*x2*x3*x4*... where all the x's are members of H.
The set H is not usually unique. (Any subset of elements {a,b,c,...} of G similarly generates a group which must be a subgroup of G.) The order of the smallest possible such set H may be though of as a characterstic "dimension" of the group, analogous to the 1,2,3 dimensions of a point, line, plane in Euclidean space, i.e. as the number of independent "directions" which extend outwards from the origin (identify). |
Group |
A set G = {a, b, c,...} and a binary operation *
with the following properties: (i) closure: For any a, b in G, a * b = is in the set. (ii) associativity: For all a, b, c in G, (a * b )* c = a * ( b * c ) . (iii) identify: There exists an element I such that I * a = a for every a in G. (iv) inverse: For every element a there exists an a-1 such that a * a-1 = I. | Homomophism |
A homomorphism from G into H is a map f which perserves the group operation,
i.e. for all g1, g2 in G, f(g1 g2) = f(g1) f(g2). See kernel. |
Isomorphic |
Two groups G and H are isomorphic if and only if there exists a 1-to-1 map
between them which preserves the group multiplication table. In other
words, if g1 and g2 are members of G, and h1=f(g1), h2=f(g2) are the
corresponding members of H under the 1-to-1 map f, then f(g1*g2)=f(g1)*f(g2). Intuitively, isomorphic groups are essentially the same for all practical purposes. |
Kernel |
The kernel of a homomorphism G->H is the set of elements of G which
are mapped to the identify of H. The kernel is always a normal subgroup of G, and its cosets form a quotient group G/(kernel) which is isomophic to H. See quotient group. |
Lagrange's Theorem |
The order of a subgroup H of a group G divides the order of G. (See "coset" for an outline of a proof.) |
Map | A relation between two sets X={a,b,c,...} and Y={A,B,C,...} that given any element in X specifies an element of Y. Often written as X->Y. The relation may be given explictly or (more often) by some kind of rule. Every element of X _must_ be mapped to something in Y. The converse is not necessarily true; there may be "untouched" elements in Y. X and Y may be the same. (See Function, 1-to-1). Example: X={1,2,3}, Y={1,2,3,4}, x->1 for any x in X. (As a function f, f(1)=1, f(2)=1, f(3)=1.) | Matrices |
There are several ways to define these depending on how picky you
want to get. Technically, matrices are linear maps in an N-dimensional
vector space, which can be written as a table of numbers in a given
basis for the vector space. Practically speaking, the matrix is usually just thought of
as that NxN table of numbers. For example, the 3x3 "identity" matrix looks like this:
Matrices are worthy of a whole course unto themselves - it's called "linear algebra" - in which you learn about their inverses, determinants, eigenvalues, eigenvectors, diagonalization, and a whole lot of other multi-syllabic words that we won't get into here. But the easy parts are so common and so useful in group theory that we may make some use of them. Also see determinant. |
Normal |
A subgroup J of a group G is "normal" if any of these three equivalent
conditions are met:
|
Onto | A map f:{x}->{y} such that for each element y there exists an x such that f(x)=y; i.e. the map touches every part of {y}. (See "1-to-1", "bijection".) | Orbit |
Given an element x of a group G, the orbit of x is the
set of all elements of G which are generated by x,
i.e. {x, x2, x3, ... }. For any element x, the orbit of x is a subgroup of G isorphic to CN, the cyclic group of N elements, where xN=I. |
Order | Generally speaking, "how many." More specifically, the order of a group (or subgroup) is how many elements there are in that group (or subgroup). By the order of an element of a group we usually mean how many elements there are in its orbit, i.e. the order of an element x is the smallest positive integer N such that xN=I. This is also called the period of that element. | Period | The period of a group element is another term for its order. | Normal |
A subgroup J of a group G is "normal" if any of these three equivalent
conditions are met:
|
Quotient group |
Given a group G and a normal subgroup J, the set of cosets of J
form a group G/J of order ord(G)/ord(J) whose group operation is
given by (xJ)(yJ)=(xyJ) where each () represents one of the cosets. This is also called "G mod J". |
Representation |
A representation of a group G is a set of matrices M which
are homomorphic to the group. In other words, there must
exists a map f:G->M such that f(g1 g2) = f(g1) * f(g2)
where "*" here refers to the usual matrix multiplication. Representations of groups is a whole branch of group theory unto itself. |
Rubik's Cube | A group-theory permutation puzzle made up of a 3-dimensional array of smaller "cubies" ( N3 of them, where N=2,3,4,...) with colored faces. But you already knew that... | Scalar | A single real or complex numeric value. (See "Vector", "Matrix".) | Semi-direct product | If a group G has a normal subgroup N, and thus can be factored as G/N = M, then we also say that G is the "semi-direct" product of N and M, G = N x| M. | Simple |
A simple group is one which has only two normal subgroups: the identity
element and the entire group. Simple groups cannot be factored,
and so are analogous to prime numbers.
(hard) Question: what is the smallest non-abelian simple group? |
Set | A collection of elements {a, b, c, d, ... } of any kind. May be empty. Size may be zero (null set, {}), finite (example: {1,2,3}) or infinite (example: {integers}). | Subgroup | A subset of a group which is also a group. | Vector | Technically, a member of a linear vector space with certain kinds of addition properties which can be written as a column of numbers given a specific basis for the vector space. Practically speaking, we usually just imagine the vector as a column (or row) of numbers, A = (1,0,0). (See "Scalar", "Matrix") |