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 TheoremIf 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: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: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.1 | 0 | 0 |
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Matrices are worthy of a whole course unto themselves (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 will make some use of them. Most calculus-level or even pre-calculus level textbooks discuss some of the operations you can perform with matrices, and we'll go over some of the basics of how they work and how they look from the perspective of group theory in class. See, for example, MatrixMultiplication.html or more examples. (I may have a more detailed online tutorial later.)
Here's a javascript 3x3 matrix calculator.
Also see determinant.
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. Normal A subgroup J of a group G is "normal" if any of these three equivalent conditions are met: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. See the java applet at the top of this web page. 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?
Answer: A5. (This fact is directly related to one of the major math results of the last century, namely that you can solve the general 4th order polynomial, but not the general 5th order one.)
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