A bijective mapping on a finite set S is called a permutation on S. In this post, we will discuss the order of a permutation, how to find the order of a permutation with examples, and related theorems. Definition of Order of a Permutation Order of permutation:- The order of a permutation σ on a … Read more
An isomorphism of groups is a special kind of group homomorphisms. It preserves every structure of groups. In this article, we will learn about isomorphism between groups, related theorems, and applications. Definition of Isomorphism A map Φ: (G, 0) → (G′, *) between two groups is called an isomorphism if the following conditions are satisfied: A … Read more
A group homomorphism is a map between two groups that preserves the algebraic structure of both groups. In this section, we will learn about group homomorphism, related theorems, and their applications. Definition of Group Homomorphism A map Φ: G → G′ between two groups (G, 0) and (G′, *) is called a group homomorphism if the … Read more
The kernel of a group homomorphism is an interesting subgroup of the domain group. This subgroup (kernel) determines whether it is an injective homomorphism or not. In this article, we will learn about the kernel of group homomorphisms. What is a kernel (Algebra)? Definition of a Kernel of a Homomorphism Let Φ: (G, 0) → … Read more
If a group has order p and p is a prime, then we call that group to be a group of prime order. A group of prime order has a nice description, and they can be characterized as follows: In this post, we will learn about groups of prime orders with their properties. Group of … Read more
Abelian groups are special types of groups in which commutativity holds. In other words, the binary operation on such groups is commutative. Abelian groups (also called commutative groups) are named after mathematician Niels Henrik Abel. In this article, we will discuss abelian groups with their properties. What is an Abelian Group? Definition: A group G … Read more
The first isomorphism theorem for groups proves that every homomorphic image of a group is actually a quotient group. This theorem is also known as the fundamental theorem of homomorphism. In this article, we will learn about the first isomorphism theorem for groups and the theorem is given below. First isomorphism theorem of groups: Let … Read more
A cyclic group is a special type of group generated by a single element. If the generator of a cyclic group is given, then one can write down the whole group. Cyclic groups are also known as monogenous groups. In this article, we will learn about cyclic groups. Definition of Cyclic Groups A group (G, … Read more
The order of a group and its elements are very crucial in group theory (Abstract Algebra). One can study groups by analyzing the orders of the group and their elements. In this article, we will learn about the order of groups. Order of a group The order of a group G is the cardinality of … Read more