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Isomorphic

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In mathematics, isomorphism is a concept that is very frequently used in various areas. Etymologically, the word isomorphism is taken from two Greek language words "isos" which means "equal" and "morphe" which refers to "shape". In general, isomorphism is referred to equal or similar shapes. Usually, two figures or objects are said to be isomorphic if they have similar shapes.
There are several different meanings and definitions of isomorphism in various fields of mathematics. Isomorphism is the property or virtue and the objects or the groups or the rings that possess this property are known as isomorphic.
Isomorphism is an important property in context of linear and abstract algebra. It may be defined as a one-to-one relation between the elements of two sets, such that the products or sums of the elements of one set and those of the other set are preserved.

In this page, we are going to go ahead and understand about the concept of isomorphism and isomorphic objects as well as their properties.

Definition

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In modern or abstract algebra, a group is defined as a set of elements specified with an operation which combines two elements forming the third element.
The operation satisfies four axioms: closure, identity, inverse and associativity. The isomorphism and homomorphism are two important properties of groups.

If (P , "o") and (Q , "x") are two groups and there is a mapping M : P $\rightarrow$ Q. Then, M is known as a homomorphism if
M(a o b) = M(a) x M (b) ; for every a , b $\in$ P.

If a homomorphism is bijective, then it is known as an isomorphism. Therefore, in addition to above, if M is a bijective or one-to-one mapping, then it is said be an isomorphism. The notation used for isomorphism is $\cong$. Here, P is isomorphic to Q and it is written as P $\cong$ Q.

Properties

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Following are the properties of isomorphism and isomorphic groups :

1) An isomorphism is bijective, i.e. there exists an inverse.

2) In an isomorphism group mapping f : P $\rightarrow$ Q, f(1) = 1 and also f (x$^{-1}$) = f(x)$^{-1}$.

3) The kernel of the group isomorphism f : P $\rightarrow$ Q is the set {e$_{P}$}, where, e$_{P}$ is identity of group P.

4) Two isomorphic groups must be of same order. If they are of different orders, they are not isomorphic to each other. For example, if one group is abelian and another is not, then they are not isomorphic.

5) If a group P is locally finite and is isomorphic to another group Q, then Q will also be locally finite.

Isomorphic Groups

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When a mapping is defined from one group to another and it follows isomorphism, then the two groups are known as isomorphic groups. We can say that one group is isomorphic to another. Let (G , o) and (H , *) are two groups and there is a group map
f : G $\rightarrow$ H
such that f is isomorphism. In this case, the groups G and H are called isomorphic groups. We can also say that G is isomorphic to H or H is isomorphic to G.

Isomorphism Theorem

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There are three basic theorems based on isomorphism of groups which are known as isomorphism theorems. These theorems are given below :

First Theorem:

Let us suppose that A and B are two groups and f : A $\rightarrow$ B be a homomorphism. According to first isomorphism theorem:
1) The kernel of f is normal subgroup of A.

2)
The image of f is subgroup of B and it is isomorphic to the group $\frac{G}{ker(f)}$.

i.e. B is isomorphic to $\frac{A}{ker(f)}$ if f is surjective.

Second Theorem:

Suppose that A be a group and X be a subgroup of A. Also assume that N is the normal subgroup of A. According to second isomorphism theorem:
1) The product of subgroup and normal subgroup i.e. XN will be the subgroup of A.

2) The intersection of subgroup and normal subgroup i.e. X $\cap$ N will be the normal subgroup of X.

3) The groups $\frac{XN}{N}$ and $\frac{X}{X \cup N}$ are isomorphic.

Third Theorem:
Let us consider that A be a group and N and M be the normal subgroups of A, such that
M $\subseteq$ N $\subseteq$ A. Then, third isomorphism theorem states that:

1) The quotient group $\frac{N}{M}$ is the normal subgroup of $\frac{A}{M}$.

2) The quotient group $\frac{\frac{A}{M}}{\frac{N}{M}}$ is isomorphic to $\frac{A}{N}$.

Isomorphic Graph

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In graph theory, the concept of isomorphic graphs is an important one. Two graphs are said to be isomorphic graphs if they contain same number of vertices that are connected in the similar way.

A simplest isomorphic graph is shown below :
 Isomorphic Graph

According to the more formal definition :
Let us suppose that X and Y be two graphs having graph vertices V$_{n}$ = {1, 2, ..., n}. Then, X and Y will be isomorphic if there exists a permutation P for graph vertices in such a way that the set {u , v} is the subset of graph edges of X i.e. E(X) if and only if the set {P(u) , P(v)} is the subset of graph edges of Y i.e. E(Y).

Subgraph Isomorphism

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In computer science, the subgraph isomorphism problem is very common computational problems in which two graphs X and Y are given (termed as input). In this problem, one needs to determine if X possesses a subgraph which is isomorphic to Y. The term subgraph matching is also utilized for the problem of subgroup isomorphism. Finding subgraph isomorphism is quite an important problem used in many areas in theoretical computer science dealing with data represented via graphs. There are many algorithms that have been proposed in order to solve this problem in a less time.