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Group Theory

A group is a set containing of elements that are combined together with an operation in such a way that any two of its elements combine via that operation to form a third element which also belongs to the set and satisfies the four axioms namely closure, associativity, invertibility and identity, and known as the group axioms. For example: group of integers under multiplication. Whereas geometry group theory is an area of mathematics that is devoted completely to the study of groups that are finitely generated with the use of the exploration of the connections between the algebraic properties of these groups and also topological and geometric properties of the spaces on which these groups are to act. 

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Introduction to Group Theory

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Group Axioms:
If ‘.’ is an operation and G is a group, then we define the group axioms as follows:

1)  Closure:
If ‘a’ and ‘b’ are two elements in G, then a . b will also belong to G

2)  Associativity:
If ‘a’, ‘b’ and ‘c’ are in group G, then a . (b . c) = (a . b) . c

3)  Identity:
For any element ‘a’ in G, there exist an element ‘I’ in G, such that a. I = I . a = a. ‘I’ is called the identity element of G.

4) Invertibility:
For every ‘a’ in G, there exist some ‘b’ in G such that a . b = b . a = I, where ‘I’ is the identity element of G.
The most common among all the examples of groups is the set of integers with addition operation along with the group. The addition of any two integers forms another integer, so closure property is satisfied. Also, addition of integers is associative, there also exist an identity element namely zero in the group and also for every integer there exists an inverse of it such that the result is zero. So, all the properties of the group are satisfied in case of integers group with addition operation.

Groups are into sharing of a basic kinship with symmetry notion. Likewise, a group symmetry is encoding the feature of symmetry of a geometrical object, that is, the group will consist of the transformation sets that will leave the object unchanged and the operation will be of combination of two such kind of transformations by performance one after of another.

Various notions have been devised by the mathematicians to explore the groups to break the groups into smaller and easy in understanding pieces like quotient groups, subgroups and simple groups.

Some Consequences of the Group Axioms:
Repetitive use of associative axiom of group theory has consequence for removing of parentheses as they can be used anywhere in the associative law.
A . B . C = (A . B) . C = A . (B . C)
Also, the identity element and the inverse element in a group are always unique.
We will prove it below. We assume that an element ‘a’ in a group (G, .) has two inverses namely ‘b’ and ‘c’. Then we have:
b = b . I     (I be the identity element in G)
Since, ‘c’ is inverse of ‘a’, so, a . c = I
    b = b . (a . c)
    b = (b . a ) . c
Since, ‘b’ is an inverse of ‘a’, so b . a = I
    b = I . c
Since I is the identity element, so I . c = c
    b = c
Thus, the two inverses are equal and hence the inverse of an element in a group is unique. Similarly we can also show that the identity element in a group is also unique.
It is possible to perform division as well in groups. For given elements ‘a’ and ‘b’ in group G, there exist exactly one solution ‘p’ to the equation p . a = b. similarly there exist only one solution ‘x’ in G to the equation a . x = b which is x = a-1 . b.

Proofs

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1) If G is a group in a way that we are given x, y $\in $ G, then (x ⋆ y)-1 = y-1 ⋆ x-1

Proof: To prove this we will show the following:
(x ⋆ y) ⋆ y-1 ⋆ x-1= I,         where I is the identity element of G.
Consider the left hand side of above we have,
(x ⋆ y) ⋆ y-1 ⋆ x-1 = x ⋆ (y ⋆ y-1) ⋆ x-1
 = x ⋆ I ⋆  x-1         (by associative axiom)
 = (x ⋆ I) ⋆  x-1      (by identity axiom)
 = x ⋆  x-1              (by identity axiom)
 = I                         (by identity axiom)
 = right hand side.
Thus we proved the required result.

2) If in a group G, ‘a’, ‘b’ and ‘c’ are three elements such that a ⋆ b = c ⋆ b, then a = c.

Proof: Let us assume that a ⋆ b = c ⋆ b.         (i)
Since ‘b’ is an element of group G, this implies there exist some ‘x’ in G with identity element I, such that
b ⋆ x = I              (ii)
On multiplying both sides of (i) by ‘x’ we get,
a ⋆ b ⋆ x = c ⋆ b ⋆ x
a ⋆ (b ⋆ x) = c ⋆ (b ⋆ x)            (by associativity)
a * I = c * I                                    [using (ii)]
a = c                                            (by identity axiom)
Thus the required result is proved. This is also known as cancellation law.

3) If G is a group and P and Q are subgroups of G, then P ⋂ Q is also a subgroup of G.

Proof: Since P and Q are subgroups, so there exist an identity element I such that I $\in $ P and I $\in $ Q.
This implies, I $\in $ P $\cap $ Q.
Let p, q $\in $ P $\cap $ Q. This implies p, q $\in $ P and p, q $\cap $ Q

Geometric Group Theory

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The main idea in the geometric group theory is to take in consideration the groups that are finitely generated as geometric objects. This is generally done by the study of the Cayley graphs of the groups which are endowed with the structure of the metric spaces in addition to the structure of the graph, which is given by the word so – called as metric.

Geometric group theory is a differentiated area and is comparatively new. It became a branch of mathematics that was clearly identifiable in the late 1980s and early period of 1990s. The geometric group theory interacts very closely with the low dimensional topology, algebraic topology, hyperbolic geometry, differential geometry and computational group theory. There also exist substantial connections of the theory with complexity theory, the study of Lie groups, mathematical logic and also their differentiated subgroups along with dynamical systems, K – theory, probability theory and various other areas of the mathematics.
Some Modern Themes & Developments:

•    The program of Gromov that studies quasi – isometric properties of the groups.

•    The theory based on word – hyperbolic and of relatively groups that are hyperbolic.

•    Interactions with the mathematical logic along with the study of the free group theory of the first order.

•    Interactions with the computer science, the theory of the formal languages and the complexity theory.

•    The study of the Dehn functions, isoperimetric inequalities along with their generalizations of the groups that are finitely presented.

•    Development of the theory of JSJ decompositions that was developed for the groups that are finitely generated and presented as well.

•    Connections with the study of C* algebras, geometric analysis that are associated with the theory of free probability and of discrete groups.

•    The sub division rules that are finite and also are in relation to Cannon’s conjecture.

•    The study of the actions of the groups of CAT (0) spaces and CAT (0) complexes in cubic that are motivated by the ideas from the geometry of Alexandrov.

•     Introductions to the methods of probability implied to study the algebraic properties of the objects of group theory in random.

•    The study of the iterated monodrony groups and automata groups, as the groups of infinite rooted trees of automorphisms.

•    The study of the properties of the group actions of the type measure theoretic on measure spaces.

•    The study of the representations  of the discrete groups that are unitary along with Kazhdan’s property.

•    The study of the outer automorphism group that is of a free group that too of rank ‘n’ and also of the individual automorphisms of the free groups.

•    The development of the theory of Bass – Serre.

•    The study of the walks on groups which are random along with the related boundary theory.

•    Study of the subgroups and the lattices in groups that are linear.

•    Progress on the combinational group theory topics that are traditional like the Burnside problem, the Coxeter groups studies and studies of the Artin groups and so on.

Some of the examples that are generally studied in the geometric group theory are amenable groups, the infinite cyclic group Z, free groups, hyperbolic groups, symmetric groups, coxeter groups, Thompson’s group F, Automatic groups, arithmetic groups, wallpaper groups, Braid groups, free products, free burnside groups etc.

Symmetry 

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In Algebra, the group of symmetry of an object is the group of all the isometries under the tag of which the object is unchanging with the group operation of composition. It is also a subgroup of the group of isometry of the space that is taken in consideration.

The objects refereed to here can be geometric figures, patterns and images likewise, the pattern of wallpaper. The definition can be more precised by making the specifications of what is the meaning of pattern or image referred to here. For example: a function with values in a set of colors of the position.

For the symmetry in physical objects, one would also interest to take the physical composition into consideration. The iso-metries group of space induces on the objects of it the group action.

The group of symmetry is often also called full symmetry group so as to emphasize that is also includes the iso-metries that are orientation reversing and under which the figures are invariant. The subgroup of the iso-metries of orientation reversing type and in which the figures are invariant is termed as the proper symmetry group which is equal to its corresponding full symmetry group only when the object is chiral and hence there exist no iso-metries under which the object is invariant and orientation reversing.

Any kind of symmetry group whose elements are having a common fixed point and the condition holds true for all symmetry groups that are finite as well as for the symmetry groups of the bounded figures can also be represented by choosing a fixed point to be the origin as a subgroup of an orthogonal group O(n). Then, the proper symmetry group is a subgroup of the orthogonal group that is special, SO (n) and thus is also termed as the rotation group of the figure.
There are three types of the discrete symmetry groups:
1) Finite point groups in which are included only reflections, rotations, roto-inversion and inversion. We can say that they are     simply the finite subgroups of O (n).
2) Infinite lattice groups in which are included only the translations.
3) Infinite space groups in which combinations of elements of both of the previous types are included along with some extra transformations like glide reflection and screw axis.
There also exist continuous symmetry groups that contain the rotations of very small angles or the translations of very small distances. The example of this is the group of all symmetries of a sphere O (3). In general, these continuous symmetry groups are also known as Lie groups.

Two figures of geometry are considered to be of the same type of symmetry if their groups of symmetry are the conjugate subgroups of the isometry group of R^n or the Euclidean group E (n), in which two subgroups H1, H2 are conjugates in a group G, if and only if there exists some g

Problems

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Question 1: If G is a finite group and let us assume that for any two subgroups H and K either one of them is contained in another, that is, either H $\subseteq $ K or K $\subseteq $ H, then prove that G is a cyclic group of power order of prime.

Solution: Since we are given that G is a finite group, so there exist an element say ‘a’ of maximal order.
This implies that <a> cannot be completely contained in any other subgroup that is cyclic, so this further implies that for every b $\in $ G, <b> $\subseteq $<a>.

Thus we get that every element of G is a power of ‘a’, which means that G is a cyclic group.
Let us say G = <a>.

Let us assume that there exist two unequal prime divisors ‘x’ and ‘y’ of G. This means there exist corresponding subgroups of order ‘x’ and ‘y’ of G which cannot be contained in the other. This contradicts our hypothesis.

Question 2: Find the order of the finite group whose only automorphism is the identity map.

Solution: Since all the inner automorphisms are of trivial type hence, G is an abelian group.
Then, $\rho $(x) = x-1 is also an automorphism and hence trivial, which implies that x = x(-1) for all x $\in $ G.
If G is additively written then there exists a vector space structure of G over the field Z2.

So any group homomorphism is a linear transformation with this vector space structure, and so the automorphism group of G will be a group of invertible matrices.

Therefore we get, Aut (G) is a non trivial, until and unless G is either zero or one dimensional.
So the order of the finite group with only automorohism as identity map is at most 2.

Question 3: Given the vectors (x1, x2, x3) and (y1, y2, y3) in R3 with the cross product defined as follows:
(x1, x2, x3) X (y1, y2, y3) = (x2 y3 – a3 b2, x3 b1 – x1 b3, x1 y2 – x2 y1)
Prove that R3 is not a group under this multiplication.

Solution: Since the cross product of the zero and any other vector is equal to zero vector only, so the cross product multiplication cannot be used to form the set of all vectors that are in R to form a group.
Hence, R3 does not form a group under cross product multiplication of three dimensional vectors.

Applications of Group Theory

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Below are listed some applications of the group theory:
1)  Galois theory use the theory of groups in order to describe the symmetries that occur in roots of a polynomial. The fundamentals of this theory are providing a link in between group theory and the algebraic field extensions. It also gives an effective way for solving the polynomial equations in terms of solving their corresponding Galois group respectively. Likewise, the symmetric group S5 in 5 elements cannot be solved at all. This implies that the general equation of quintic is not solvable by the radicals just in the way we can solve the equations of lower degree. This theory is still been applied to obtain the new results in fields like class field theory affectively.

2)  Another application of group theory is algebraic topology. Algebraic topology mainly associates the groups with the objects in which the interest of theory is shown. Groups are actually used to describe some invariants that are there in topological spaces. The invariants are so called as they are defined such that they do not change even if the topological space is subjected to any deformation or change.

3) Algebraic geometry and cryptography are other fields of application of group theory in several ways. We have explained about the abelian varieties above. Here, these varieties are simple accessible due to the presence of an operation in the group that produces more information. Sometimes, they also serve as the tests for some new conjectures. Elliptic curves are studied as a case of one dimensional space that too in detail in both practical and theoretical ways. Caesar’s cipher, toric varieties are also some of the theories that are considered as group theories.

4) The algebraic number theory is a special type of group theory. Let us take the example of Euler’s product formula. This formula is based on the fact that any integer can be decomposed into primes in a unique way. but this statement failed and it gave rise to class groups and regular prime groups.
More topics in Group Theory
Direct Product of Groups
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