Abstract Algebra

Lagrange’s theorem states that the order of any subgroup of a finite group divides the order of the group. In this page, we will state and prove Lagrange’s theorem along with some applications and solved problems. Statement The Lagrange’s theorem for finite groups states the following: Let G be a finite group and H be … Read more

In this page, we prove that every subgroup of cyclic group is Cyclic. Every subgroup of a cyclic group is cyclic Proof: Let G be a cyclic group. By definition, there exists an element g ∈ G such that G = <g> = {gk : k∈$\mathbb{Z}$ }. Let H be a subgroup of G. We … Read more

In this page, we prove that every group of prime order is cyclic. Prove that Group of Prime Order is Cyclic Answer: Let G be a group of order p where p is a prime number So |G| = p. As p is a prime number, p>1. That is, |G| > 1. So there is … Read more

In this page, you can find a list of Abstract Algebra practice problems. You need to submit the assignment based on the problems below. Group Theory Q1: Define semigroup and monoid with examples. Q2: Is the set of natural numbers form a group under addition? Give reasons. Q3: Let G = {2n: 𝑛 ∈ 𝑍}. … Read more