The orbit-stabilizer theorem of groups says that the size of a finite group G is the multiplication of the size of the orbit of an element a (in A on which G acts) with that of the stabilizer of a. In this article, we will learn about what are orbits and stabilizers. We will also … Read more
In this article, we will learn about the definition of quotient groups along with their examples, properties, and a few solved problems. Definition of Quotient Group Let H be a normal subgroup of a group G. Consider the set of all left cosets of H in G, that is, {aH: a∈G}. This set forms a … Read more
The center of a group G is a subset containing those elements of G that commute with every element of the group G. In this article, we will learn about the center of a group and show that it is a normal subgroup. Center of a Group Definition Let (G, o) be a group. Then … Read more
A semigroup in mathematics is a set equipped with a binary operation that is associative. In this article, we will study the definition of semigroups together with examples, and properties. Definition of a Semigroup Let G be a non-empty set and o be an algebraic operation acting on it. Then the pair (G, o) is … Read more
In Group theory, we analyze the algebraic structures of a set with a binary operation given. In this article, we will learn the definition of a group (in Abstract Algebra) with their properties, examples, and applications. Definition of a Group Let G be a set and o be a binary operation acting on it. Then … Read more
A simple group is basically a group having no proper nontrivial normal subgroups. For example, A5 is a simple group. In this post, we will learn about simple groups with examples, properties, and classification. Definition of Simple Group A group is called a simple group if its only normal subgroups are the trivial subgroup and … Read more
A normal subgroup H of a group G is a subgroup of G that is invariant under conjugation by members of the group. In other words, every left coset and right coset corresponding to an element g are the same, that is, gH=Hg. Normal subgroups have many applications. In this post, we will learn about normal subgroups with … Read more
Cosets are mainly used to decompose a group G into equal-sized disjoint subsets of G. It plays an important role to study many things in Group Theory; for example, normal group, Lagrange’s theorem on finite groups, etc. In this post, we will learn about cosets, their classification with examples, and their properties with related theorems. … Read more
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
