Tag Archives: group



Pentominoes: Puzzles & Solutions

PENTOMINOES – An Introduction

A pentomino is a plane geometric figure formed by joining five equal squares edge to edge. It is a polyomino with five cells. There are twelve pentominoes, not counting rotations and reflections as distinct. They are used chiefly in recreational mathematics for puzzles and problems.[1] Pentominoes were formally defined by American professor Solomon W. Golomb starting in 1953 and later in his 1965 book Polyominoes: Puzzles, Patterns, Problems, and Packings.[1][2] Golomb coined the term “pentomino” from the Ancient Greek πέντε / pénte, “five”, and the -omino of domino, fancifully interpreting the “d-” of “domino” as if it were a form of the Greek prefix “di-” (two). Golomb named the 12 free pentominoes after letters of the Latin alphabet that they resemble.

Ordinarily, the pentomino obtained by reflecting or rotating a pentomino does not count as a different pentomino. The F, L, N, P, Y, and Z pentominoes are chiral; adding their reflections (F’, J, N’, Q, Y’, S) brings the number of one-sided pentominoes to 18. Pentominoes I, T, U, V, W, and X, remain the same when reflected. This matters in some video games in which the pieces may not be reflected, such as Tetris imitations and Rampart.

Each of the twelve pentominoes satisfies the Conway criterion; hence every pentomino is capable of tiling the plane.[3] Each chiral pentomino can tile the plane without reflecting it.[4]

John Horton Conway proposed an alternate labeling scheme for pentominoes, using O instead of I, Q instead of L, R instead of F, and S instead of N. The resemblance to the letters is more strained, especially for the O pentomino, but this scheme has the advantage of using 12 consecutive letters of the alphabet. It is used by convention in discussing Conway’s Game of Life, where, for example, one speaks of the R-pentomino instead of the F-pentomino.

a handout given to physics students at Harvard

Common knowledge is a special kind of knowledge for a group of agents. There is common knowledge of p in a group of agents G when all the agents in G know p, they all know that they know p, they all know that they all know that they know p, and so on ad infinitum.[1]

The concept was first introduced in the philosophical literature by David Kellogg Lewis in his study Convention (1969). The sociologist Morris Friedell defined common knowledge in a 1969 paper.[2] It was first given a mathematical formulation in a set-theoretical framework by Robert Aumann (1976). Computer scientists grew an interest in the subject ofepistemic logic in general – and of common knowledge in particular – starting in the 1980s.[1] There are numerous puzzles based upon the concept which have been extensively investigated by mathematicians such as John Conway.[3]

The philosopher Stephen Schiffer, in his book Meaning, independently developed a notion he called “mutual knowledge” which functions quite similarly to Lewis’s “common knowledge”.[4]

Green-eyed dragons

You visit a remote desert island inhabited by one hundred very friendly dragons,
all of whom have green eyes. They haven’t seen a human for many centuries and
are very excited about your visit. They show you around their island and tell you
all about their dragon way of life (dragons can talk, of course).
They seem to be quite normal, as far as dragons go, but then you find out
something rather odd. They have a rule on the island which states that if a dragon
ever finds out that he/she has green eyes, then at precisely midnight on the day of
this discovery, he/she must relinquish all dragon powers and transform into a longtailed
sparrow. However, there are no mirrors on the island, and they never talk
about eye color, so the dragons have been living in blissful ignorance throughout
the ages.
Upon your departure, all the dragons get together to see you off, and in a tearful
farewell you thank them for being such hospitable dragons. Then you decide to tell
them something that they all already know (for each can see the colors of the eyes of
the other dragons). You tell them all that at least one of them has green eyes. Then
you leave, not thinking of the consequences (if any). Assuming that the dragons are
(of course) infallibly logical, what happens?
If something interesting does happen, what exactly is the new information that
you gave the dragons?

Linear algebraic group

Linear algebraic group

From Wikipedia, the free encyclopedia
In mathematics, a linear algebraic group is a subgroup of the group of invertible n×n matrices
(under matrix multiplication) that is defined by polynomial equations.

An example is the orthogonal group, defined by the relation MTM = I where MT is the transpose of


In abstract algebra, an algebraic structure consists of one or more sets, called underlying sets or carriers or sortsclosed under one or more operations, satisfying some axioms. Abstract algebra is primarily the study of algebraic structures and their properties. The notion of algebraic structure has been formalized in universal algebra.

Types of magmas

Magmas are not often studied as such; instead there are several different kinds of magmas, depending on what axioms one might require of the operation. Commonly studied types of magmas include
Magma to Group.svg

From magma to group, via two alternative paths. Key:
M = magma, d = divisibility, a = associativity,
Q = quasigroup, S = semigroup, e = identity.
L = loop, i = invertibility, N = monoid, G = group
Note that both divisibility and invertibility imply
the existence of the cancellation property.

Noether’s (first) theorem

Noether’s (first) theorem states that any differentiable symmetry of the action of a physical system has a corresponding conservation law. The theorem was proved by German mathematician Emmy Noether in 1915 and published in 1918.[1] The action of a physical system is the integral over time of a Lagrangian function (which may or may not be an integral over space of a Lagrangian density function), from which the system’s behavior can be determined by the principle of least action.

Noether’s theorem has become a fundamental tool of modern theoretical physics and the calculus of variations. A generalization of the seminal formulations on constants of motion in Lagrangian and Hamiltonian mechanics (1788 and 1833, respectively), it does not apply to systems that cannot be modeled with a Lagrangian; for example, dissipative systems with continuous symmetries need not have a corresponding conservation law.

For illustration, if a physical system behaves the same regardless of how it is oriented in space, its Lagrangian is rotationally symmetric; from this symmetry, Noether’s theorem shows the angular momentum of the system must be conserved. The physical system itself need not be symmetric; a jagged asteroid tumbling in space conserves angular momentum despite its asymmetry – it is the laws of motion that are symmetric. As another example, if a physical experiment has the same outcome regardless of place or time (having the same outcome, say, somewhere in Asia on a Tuesday or in America on a Wednesday), then its Lagrangian is symmetric under continuous translations in space and time; by Noether’s theorem, these symmetries account for the conservation laws of linear momentum and energy within this system, respectively. (These examples are just for illustration; in the first one, Noether’s theorem added nothing new – the results were known to follow from Lagrange’s equations and from Hamilton’s equations.)

Noether’s theorem is important, both because of the insight it gives into conservation laws, and also as a practical calculational tool. It allows researchers to determine the conserved quantities from the observed symmetries of a physical system. Conversely, it allows researchers to consider whole classes of hypothetical Lagrangians to describe a physical system. For illustration, suppose that a new field is discovered that conserves a quantity X. Using Noether’s theorem, the types of Lagrangians that conserve X because of a continuous symmetry can be determined, and then their fitness judged by other criteria.

There are numerous different versions of Noether’s theorem, with varying degrees of generality. The original version only applied to ordinary differential equations (particles) and not partial differential equations (fields). The original versions also assume that the Lagrangian only depends upon the first derivative, while later versions generalize the theorem to Lagrangians depending on the nth derivative. There is also a quantum version of this theorem, known as the Ward–Takahashi identity. Generalizations of Noether’s theorem to superspaces also exist.


Symmetry in Science: An Introduction to the General Theory






Special unitary group

Rubik's cube v2.svg

In mathematics, the special unitary group of degree n, denoted SU(n), is the group of n×n unitary matrices with determinant 1. The group operation is that ofmatrix multiplication. The special unitary group is a subgroup of the unitary group U(n), consisting of all n×n unitary matrices, which is itself a subgroup of thegeneral linear group GL(nC).

The SU(n) groups find wide application in the Standard Model of particle physics, especially SU(2) in the electroweak interaction and SU(3) in QCD.

The simplest case, SU(1), is the trivial group, having only a single element. The group SU(2) is isomorphic to the group of quaternions of absolute value 1, and is thus diffeomorphic to the 3-sphere. Since unit quaternions can be used to represent rotations in 3-dimensional space (up to sign), we have a surjectivehomomorphism from SU(2) to the rotation group SO(3) whose kernel is { + I, − I}.


A group expresses the intuitive concept of symmetry.

group (G, •) is a set G closed under a binary operation • satisfying the following 3 axioms:

  • Associativity: For all ab and c in G, (a • b) • ca • (b • c).
  • Identity element: There exists an eG such that for all a in Ge • aa • ea.
  • Inverse element: For each a in G, there is an element b in G such that a • bb • ae, where e is an identity element.

Basic definitions

subset H ⊂ G is a subgroup if the restriction of • to H is a group operation on H. It is called normal, if left and right cosets agree, i.e. gHHg for all g in G.

Given a subset S of a group G, the smallest subgroup of G containing S is called the subgroup generated by S. It is often denoted <S>.

Both subgroups and normal subgroups of a given group form a complete lattice under inclusion of subsets; this property and some related results are described by the lattice theorem.

group homomorphism is a map fG → H between two groups that preserves the structure imposed by the operation, i.e.

f(ab) = f(a) • f(b).

Bijective (in-surjective) maps are isomorphisms of groups (mono-epimorphisms, respectively). The kernel ker(f) is always a normal subgroup of the group. For f as above, the fundamental theorem on homomorphisms relates the structure of G and H, and of the kernel and image of the homomorphism, namely

G / ker(f) ≅ im(f).

One of the fundamental problems of group theory is the classification of groups up to isomorphism.

Groups together with group homomorphisms form a category.

Finiteness conditions

The order |G| (or o(G)) of a group is the cardinality of G. If the order |G| is (in-)finite, then G itself is called (in-)finite. An important class is the group of permutations or symmetric groups of N letters, denoted SNCayley’s theorem exhibits any finite group G as a subgroup of the symmetric group on G.

Abelian groups

The category of groups can be subdivided in several ways. A particularly well-understood class of groups are the so-called abelian (in honor of Niels Abel, or commutative) groups, i.e. the ones satisfying

a • bb • a for all ab in G.

Another way of saying this is that the commutator

[ab] := a−1b−1ab

equals the identity element. A non-abelian group is a group that is not abelian. Even more particular, cyclic groups are the groups generated by a single element. Being either isomorphic to Z or to Zn, the integers modulo n, they are always abelian.

General linear group

In mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, and the inverse of an invertible matrix is invertible. The name is because the columns of an invertible matrix are linearly independent, hence the vectors/points they define are in general linear position, and matrices in the general linear group take points in general linear position to points in general linear position.

To be more precise, it is necessary to specify what kind of objects may appear in the entries of the matrix. For example, the general linear group over R (the set of real numbers) is the group of n×n invertible matrices of real numbers, and is denoted by GLn(R) or GL(nR).

More generally, the general linear group of degree n over any field F (such as the complex numbers), or a ring R (such as the ring of integers), is the set of n×ninvertible matrices with entries from F (or R), again with matrix multiplication as the group operation.[1] Typical notation is GLn(F) or GL(nF), or simply GL(n) if the field is understood.

More generally still, the general linear group of a vector space GL(V) is the abstract automorphism group, not necessarily written as matrices.

The special linear group, written SL(nF) or SLn(F), is the subgroup of GL(nF) consisting of matrices with a determinant of 1.

The group GL(nF) and its subgroups are often called linear groups or matrix groups (the abstract group GL(V) is a linear group but not a matrix group). These groups are important in the theory of group representations, and also arise in the study of spatial symmetries and symmetries of vector spaces in general, as well as the study of polynomials. The modular group may be realised as a quotient of the special linear group SL(2, Z).

If n ≥ 2, then the group GL(nF) is not abelian.


\operatorname{SU}_2(\mathbb{C}) and \mathfrak{su}_2(\mathbb{C})

A general matrix element of \operatorname{SU}_2(\mathbb{C}) takes the form

U =  \begin{pmatrix} \alpha&-\overline{\beta}\\ \beta&\overline{\alpha} \end{pmatrix}

where \alpha,\beta\in\mathbb{C} such that | α | 2 + | β | 2 = 1. We can consider the following map \varphi : \mathbb{C}^2 \to \operatorname{M}(2,\mathbb{C}), (where \operatorname{M}(2,\mathbb{C}) denotes the set of 2 by 2 complex matrices), defined in the obvious way by

 \varphi(\alpha,\beta) = \begin{pmatrix} \alpha&-\overline{\beta}\\ \beta&\overline{\alpha} \end{pmatrix}.

By considering \mathbb{C}^2 diffeomorphic to \mathbb{R}^4 and \operatorname{M}(2,\mathbb{C}) diffeomorphic to \mathbb{R}^8 we can see that \varphi is an injective real linear map and hence an embedding. Now considering the restriction of \varphi to the 3-sphere, denoted S3, we can see that this is an embedding of the 3-sphere onto a compact submanifold of \operatorname{M}(2,\mathbb{C}). However it is also clear that \varphi(S^3) = \operatorname{SU}_2(\mathbb{C}), which as a manifold is diffeomorphic to \operatorname{SU}_2(\mathbb{C}), making \operatorname{SU}_2(\mathbb{C}) a compact, connected Lie group.

Now considering the Lie algebra \mathfrak{su}_2(\mathbb{C}), a general element takes the form

 U' =  \begin{pmatrix} ix & -\overline{\beta}\\ \beta & -ix \end{pmatrix}

where x \in \mathbb{R} and \beta \in \mathbb{C}. It is easily verified that matrices of this form have trace zero and are antihermitian. The Lie algebra is then generated by the following matrices

 u_1 = \begin{pmatrix} 0 & i\\ i & 0 \end{pmatrix} \qquad u_2 = \begin{pmatrix} 0 & -1\\ 1 & 0 \end{pmatrix} \qquad u_3 = \begin{pmatrix} i & 0\\ 0 & -i \end{pmatrix}

which are easily seen to have the form of the general element specified above. These satisfy the relations u3u2 = − u2u3 = − u1 and u2u1 = − u1u2 = − u3. The commutator bracket is therefore specified by

 [u_3,u_1]=2u_2, \qquad [u_1,u_2] = 2u_3, \qquad [u_2,u_3] = 2u_1.

The above generators are related to the Pauli matrices by u1iσ1u2 = − iσ2 and u3iσ3.


The generators of \mathfrak{su}(3), T, in the defining representation, are:

T_a = \frac{\lambda_a }{2}.\,

where \lambda \,, the Gell-Mann matrices, are the SU(3) analog of the Pauli matrices for SU(2):

\lambda_1 = \begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} \lambda_2 = \begin{pmatrix} 0 & -i & 0 \\ i & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} \lambda_3 = \begin{pmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 0 \end{pmatrix}
\lambda_4 = \begin{pmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{pmatrix} \lambda_5 = \begin{pmatrix} 0 & 0 & -i \\ 0 & 0 & 0 \\ i & 0 & 0 \end{pmatrix} \lambda_6 = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix}
\lambda_7 = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & -i \\ 0 & i & 0 \end{pmatrix} \lambda_8 = \frac{1}{\sqrt{3}} \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -2 \end{pmatrix}.

Note that they are all traceless Hermitian matrices as required.

These obey the relations

\left[T_a, T_b \right] = i \sum_{c=1}^8{f_{abc} T_c} \,

where the f are the structure constants, as previously defined, and have values given by

f_{123} = 1 \,
f_{147} = -f_{156} = f_{246} = f_{257} = f_{345} = -f_{367} = \frac{1}{2} \,
f_{458} = f_{678} = \frac{\sqrt{3}}{2}, \,

and all other fabc not related to these by permutation are zero.

The d take the values:

d_{118} = d_{228} = d_{338} = -d_{888} = \frac{1}{\sqrt{3}} \,
d_{448} = d_{558} = d_{668} = d_{778} = -\frac{1}{2\sqrt{3}} \,
d_{146} = d_{157} = -d_{247} = d_{256} = d_{344} = d_{355} = -d_{366} = -d_{377} = \frac{1}{2}. \,



Quarks and Leptons: An Introductory Course in Modern Particle Physics