On this page, we will learn about the third isomorphism theorem for groups along with its statement and proof. Third Isomorphism Theorem Statement Let G be a group and H, K be its two normal subgroups such that H ≤ K. Then we have a group isomorphism (G/H)/(K/H) ≅ G/K. Third Isomorphism Theorem Proof First, … Read more
In this article, we will learn about the second isomorphism theorem for groups. This is also known as the diamond isomorphism theorem. The statement with its proof is provided below. Second Isomorphism Theorem Statement Let G be a group such that Then we have a group isomorphism H/(H∩K) ≅ HK/K. Second Isomorphism Theorem Proof The … Read more
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