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• A common foundation for the theory of equations on the basis of the group of permutations was found by
mathematician Lagrange and on this was built the theory of substitutions.
• He discovered that the roots of all resolvents which he examined are rational functions of the roots of the
respective equations.
• To study the properties of these functions he invented a Calcul des Combinaisons. The contemporary work of
Vandermonde also foreshadowed the coming theory.
• Groups in the 1870-1900 period were described as the continuous groups of Lie, the discontinuous groups,
finite groups of substitutions of roots (gradually being called permutations), and finite groups of linear
substitutions (usually of finite fields).
• During the 1880-1920 period, groups described by presentations came into a life of their own through the
work of Arthur Cayley, Walther von Dyck, Dehn, Nielsen, Schreier, and continued in the 1920-1940 period
with the work of Coxeter, Magnus, and others to form the field of combinatorial group theory.
Fundamental groups and reflection groups encouraged the developments of J. A.
Todd and Coxeter, such as the Todd– Coxeter algorithm in combinatorial group
theory. Algebraic groups, defined as solutions of polynomial equations
(rather than acting on them, as in the earlier century), benefited heavily from the
continuous theory of
Lie. Neumann and Neumann produced their study of varieties of groups, groups
defined by group theoretic equations rather than polynomial ones.Continuous
groups also had explosive growth in the 1900-1940 period. Topological groups
began to be studied as such. There were many great achievements in continuous
groupsBoth depth, breadth and also the impact of group theory subsequently
grew. The domain started branching out into areas such as algebraic groups, group
extensions, and representation theory. Starting in the 1950s, in a huge
collaborative effort, group theorists succeeded to classify all finite simple groups in
1982. Completing and simplifying the proof of the classification are areas of active
• The last twenty years of the 20th century enjoyed the successes of over one hundred years of study in group theory .
• In finite groups, post classification results included the O'Nan–Scott theorem, the Aschbacher classification, the
classification of multiply transitive finite groups, the determination of the maximal subgroups of the simple groups and
the corresponding classifications of primitive groups. In finite geometry and combinatorics, many problems could now be
• In discrete groups, several areas of geometry came together to produce exciting new fields. Work on knot theory,
orbifolds, hyperbolic manifolds, and groups acting on trees (the Bass–Serre theory), much enlivened the study of
hyperbolic groups, automatic groups.
• Continuous groups saw the solution of the problem of hearing the shape of a drum using symmetry groups of the
laplacian operator. Continuous techniques were applied to many aspects of group theory using function spaces and
quantum groups. Many 18th and 19th century problems are now revisited in this more general setting, and many questions
in the theory of the representations of groups have answers.
Branches of group theory
• In abstract algebra, a finite group is a mathematical group
with a finite number of elements.
• A group is a set of elements together with an operation
which associates, to each ordered pair of elements, an
element of the set .With a finite group, the set is finite.
Geometric group theory
Invented in 1974 by Hungarian sculptor and architecture professorErno Rubik.
Cube: 2x2x2
 The Pocket Cube (also known as the Mini Cube or the Ice Cube) is the 2×2×2 equivalent of
a Rubik's Cube. The cube consists of 8 pieces, all corners.
 The number of possible positions of the cube is
 The maximum number of turns required to solve the cube is up to 11 full turns, or up to 14
quarter turns.
 For the miniature (2 × 2 × 2) Rubik’s cube, the two-generator subgroup (the number of
positions generated just by rotations of two adjacent faces) is of order 29,160. [2] The two
algorithms can be used to solve the last layer:
1.To fix the positions of the corners:(U R U' L' U R' U' L)
2.To adjust the corners:(D R' D' R)
 The Rubik's Revenge (also known as the Master Cube) is a 4×4×4 version of Rubik's Cube. It was
released in 1981.
 Invented by Péter Sebestény, the Rubik's Revenge was nearly called the Sebestény Cube until a
somewhat last-minute decision changed the puzzle's name to attract fans of the original Rubik's
 Unlike the original puzzle (and the 5×5×5 cube), it has no fixed facets: the centre facets (four per
face) are free to move to different positions.
 This gives a total number of permutations of
 The full number is 7401196841564901869874093974498574336000000000 possible permutations
(about 7401 septillion or 7.4 septilliard on the long scale or 7.4 quattuordecillion on the short scale).
Cube: 5x5x5
 The Professor's Cube is a mechanical puzzle, a 5×5×5 version of the Rubik's Cube.
 It has qualities in common with both the original 3×3×3 Rubik's Cube and the 4×4×4Rubik's
Revenge, and knowing how to solve either can help when working on the 5×5×5 cube.
 There are 54 centers. Six of these (the center square of each face) are fixed in position. The rest
consist of two sets of 24 centers. Within each set there are four centers of each color.
 Each set can be arranged in 24! different ways. Assuming that the four centers of each color in each
set are indistinguishable, the number of permutations of each set is reduced to 24!/(4!6)
arrangements, all of which are possible.
 This gives a total number of permutations of
 The full number is precisely 282 870 942 277 741 856 536 180 333 107 150 328 293 127 731 985 672 134
721 536 000 000 000 000 000 possible permutations[7] (about 283 duo decillion on the long scale or 283
tresvigintillion on the short scale).
 The Pyraminx is a regular tetrahedron puzzle in the style of Rubik's Cube. It
was made and patented by Uwe Mèffert after the original 3 layered Rubik's
Cube by Erno Rubik, and introduced by Tomy Toys of Japan (then the 3rd
largest toy company in the world) in
 1981.The maximum number of twists required to solve the Pyraminx is 11.
There are 933,120 different positions (disregarding rotation of the trivial tips), a
number that is sufficiently small to allow a computer search for optimal
 The current world record for a single solve of the Pyraminx stands at 1.32
solutions. seconds, set by Drew Brads at the Lexington Fall 2015.[2] The best
average time of 2.52 seconds is held by Drew Brads done at Lexington Fall
 The Megaminx is a dodecahedron-shaped puzzle
similar to the Rubik's Cube. It has a total of 50 movable
pieces to rearrange, compared to the 20 movable pieces of
the Rubik's Cube.
 There are 30!/2 ways to arrange the edges and 229 ways to flip them.
 The full number is 100 669 616 553 523 347 122 516 032 313 645 505 168 688 116 411 019 768 627
200 000 000 000 (roughly 101 unvigintillion on the short scale or 101 undecillionth on the long
The variations of possible cube configurations have inspired mathematicians to
study Rubik's cube.
However, the most common type of cube study focuses on
permutations and algorithms.
Permutation is the ordering of a series of elements, in this case ,
the ordering of the faces of the cube.
An algorithm is a step-by-step guide that outlines the
procedure for a task.
Solving Rubik's cube by reading step-by-step instructions is too easy. It takes the
challenge out of the cube, and with it, the fun.
Instead of listing out steps to solve the puzzle, try these three different moves that help
in solving the puzzle by switching or adjusting different pieces.
Adjusting a piece means to rotate the direction a color faces on a piece Switching
pieces refers to swapping the position of two different pieces A row is one of the three
horizontal divisions of the cube. A column is one of the three vertical divisions of the
 Move #1 - a move to adjust two consecutive edge pieces (alters nothing else)
 Move #2 - a move to adjust three corners of a face (alters adjacent edge pieces)
 Move #3 -- a move to switch the positions of three edges on the same plane (alters
nothing else)
 Rami Sbahi (USA) holds the current world record for solving the Pocket Cube in competition, with a time of
0.58 seconds at the Canadian Open 2015.
 The best average of five consecutive solves in competition is 1.51 seconds, set by Lucas Etter (USA) at the
Music City Speedsolving 2015 competition. The times in his average, of which the best and worst are dropped,
were (1.24), 1.69, (2.21), 1.45, and 1.39.
 Single time: The current world record for single time on a 3×3×3 Rubik's Cube was set by Lucas Etter of the
United States in November 2015 with a time of 4.90 seconds at the River Hill Fall 2015 competition in
Clarksville, Maryland.[58]
 Average time: The world record for average time per solve was set by Feliks Zemdegs at the Melbourne Cube
Day 2013, with a 6.54 second average solve time.
 One-handed solving: The fastest single time of solving the cube with one hand is 6.88 seconds by Feliks
Zemdegs at the Canberra Autumn 2015. The fastest average time of 10.87 seconds was made by Antoine
Cantin at the Toronto Spring 2015.
 Feet solving: Jakub Kipa solved a Rubik's Cube with his feet in 20.57 seconds at the Radomsko Cube Theory
2015. Gabriel Pereira Campanha got an average of 3 solves of 28.41 at the SESC Santos 2015.
 Blindfold solving: The record for blind solving is held by Kaijun Lin (林恺俊) of China, who solved a cube
blindfolded in 21.05 seconds (including memorization) at China Championship 2015.[65]
 Multiple blindfold solving: The record is held by Marcin Kowalczyk of Poland, who successfully solved 41 of
41 cubes blindfolded at the SLS Swierklany 2013.
 Fewest moves solving: Tim Wong of the United States holds the record of 19 moves set at the Irvine Fall 2015.
 Non-human solving: The fastest non-human time for a physical 3×3×3 Rubik's Cube is 3.25 seconds, set by
CubeStormer III, a robot built using Lego Mindstorms and a Samsung Galaxy S4.[66] This beats the prior 5.27
seconds, set by CubeStormer II, a robot built using Lego Mindstorms and a Samsung Galaxy S2.
• 1. Applications of group theory
abound. Almost all structures
in abstract algebra are special cases
of groups. Rings, for example, can be
viewed as abelian
groups (corresponding to addition)
together with a second operation
(corresponding to multiplication).
Therefore, group theoretic
arguments underlie large parts of the
theory of those entities.
Galois theory uses groups to describe the symmetries of the roots of a
polynomial (or more precisely the automorphisms of the algebras
generated by these roots). The fundamental theorem of Galois theory
provides a link between algebraic field extensions and group theory. It
gives an effective criterion for the solvability of polynomial equations
in terms of the solvability of the corresponding Galois group. For
example,S5, the symmetric group in 5 elements, is not solvable which
implies that the general equation cannot be solved by radicals in the
way equations of lower degree can. The theory, being one of the
historical roots of group theory, is still fruitfully applied to yield new
results in areas such as class field theory.
Algebraic topology is another domain which
prominently associates groups to the objects the
theory is interested in. There, groups are used to
describe certain invariants of topological spaces. They
are called "invariants" because they are defined in
such a way that they do not change if the space is
subjected to some deformation. For example, the
fundamental group "counts" how many paths in the
space are essentially different. The Poincare
conjecture, proved in 2002/2003 by Grigori Perelman,
is a prominent application of this idea. The influence is
not unidirectional, though. For example, algebraic
topology makes use of Eilenberg–MacLane spaces
which are spaces with prescribed homotopy groups.
Similarly algebraic K-theory relies in a way on
classifying spaces of groups. Finally, the name of the
torsion subgroup of an infinite group shows the legacy
of topology in group theory.
Algebraic geometry and cryptography likewise uses group theory in many
ways. AbeliaN varieties have been introduced above. The presence of the
group operation yields additional information which makes these varieties
particularly accessible. They also often serve as a test for new
conjectures. The one-dimensional case, namely elliptic curves is studied in
particular detail. They are both theoretically and practically intriguing.Very
large groups of prime order constructed in Elliptic-Curve Cryptography serve
for public key cryptography. Cryptographical methods of this kind benefit
from the flexibility of the geometric objects, hence their group structures,
together with the complicated structure of these groups, which make
the discrete logarithm very hard to calculate. One of the earliest encryption
protocols, Caesar's cipher, may also be interpreted as a (very easy) group
operation. In another direction, toric varieties are algebrai varieties acted on
by a torus. Toroidal embeddings have recently led to advances in algebraic
geometry, in particular resolution of singularities.
Algebraic number theory is a special case of group theory, thereby following the rules of the latter. For example,Euler's
product formula
captures the fact that any integer decomposes in a unique way into primes. The failure of this statement for more general
rings gives rise to class groups and regular primes, which feature in Kummer's treatment of Fermat's Last Theorem.
Analysis on Lie groups and certain other groups is called harmonic analysis. Haar measures, that is, integrals invariant under
the translation in a Lie group, are used for pattern recognition and other image processing techniques
In combinatorics, the notion of permutation group and the concept of group action are often used to simplify the counting of
a set of objects; see in particular Burnside's lemma
The presence of the 12-periodicity in the circle of fifths yields applications of elementary group theory in musical set theory.
Group theory can be used to resolve the incompleteness of the statistical interpretations of mechanics developed by
Willard Gibbs, relating to the summing of an infinite number of probabilities to yield a meaningful solution
In physics, groups are important
because they describe the symmetries
which the laws of physics seem to obey.
According to Noether's theorem, every
continuous symmetry of a physical
system corresponds to a conservation
law of the system. Physicists are very
interested in group representations,
especially of Lie groups, since these
representations often point the way to
the "possible" physical theories.
Examples of the use of groups in
physics include the Standard Model,
gauge theory, the Lorentz group, and
the Poincaré group.
In chemistry and materials science, groups are used to classify
crystal structures, regular polyhedra, and the symmetries of
molecules. The assigned point groups can then be used to
determine physical properties (such as polarity and chirality),
spectroscopic properties (particularly useful for Raman
spectroscopy and infrared spectroscopy), and to construct
molecular orbitals.
Molecular symmetry is responsible for many physical and
spectroscopic properties of compounds and provides relevant
information about how chemical reactions occur. In order to assign
a point group for any given molecule, it is necessary to find the set
of symmetry operations present on it. The symmetry operation is
an action, such as a rotation around an axis or a reflection through
a mirror plane. In other words, it is an operation that moves the
molecule such that it is indistinguishable from the original
configuration. In group theory, the rotation axes and mirror planes
are called "symmetry elements". These elements can be a point,
line or plane with respect to which the symmetry operation is
carried out. The symmetry operations of a molecule determine the
specific point group for this molecule.
In chemistry, there are five important symmetry operations. The
identity operation (E) consists of leaving the molecule as it is.
This is equivalent to any number of full rotations around any axis.
This is a symmetry of all molecules, whereas the symmetry group
of a chiral molecule consists of only the identity operation.
Rotation around an axis (Cn) consists of rotating the molecule
around a specific axis by a specific angle. For example, if a water
molecule rotates 180° around the axis that passes through the
oxygen atom and between the hydrogen atoms, it is in the same
configuration as it started. In this case, n = 2, since applying it
twice produces the identity operation. Other symmetry operations
are: reflection, inversion and improper rotation (rotation followed
by reflection).