Table of Contents

## What is the smallest non Abelian group order?

Non-abelian groups are pervasive in mathematics and physics. One of the simplest examples of a non-abelian group is the dihedral group of order 6. It is the smallest finite non-abelian group.

### What is the smallest order of an Abelian group which is not cyclic?

The simplest possible example of this would be Z/2Z×Z/2Z, as this is abelian and is the smallest group which is not cyclic.

#### What is the smallest Abelian group?

Smallest abelian non-cyclic group is klien four group . It has element and each non-identity element has order , hence it is non-cyclic.

**What are the possible orders for non Abelian simple groups?**

There are only 491 possible orders for non-Abelian simple groups of order less than 10 billion.

**What is the order of D3?**

D3 has one subgroup of order 3: <ρ1> = <ρ2>. It has three subgroups of order 2: <τ1>, <τ2>, and <τ3>.

## What is the order of A5?

(e) List all possible orders of an element of A5. 1, 2, 3, 5. The elements of A5 have one of the following forms: the identity, two 2-cycles, a 3-cycle, and a 5-cycle. The orders of elements of these forms, in order, are 1, 2, 3, and 5.

### Is a non Abelian group cyclic?

All cyclic groups are Abelian, but an Abelian group is not necessarily cyclic. All subgroups of an Abelian group are normal. In an Abelian group, each element is in a conjugacy class by itself, and the character table involves powers of a single element known as a group generator.

#### Can a non Abelian group have a non Abelian subgroup?

Every non-abelian group has a non-trivial abelian subgroup: Let G be a nonabelian group and x∈G, x not the identity. For example, S3 is a nonabelian group such that only the cyclic subgroups are abelian.

**Is group of order 24 Simple?**

Theorem 3.1 There is no simple group of order 24.

**What is the order of a group?**

The Order of a group (G) is the number of elements present in that group, i.e it’s cardinality. It is denoted by |G|. Order of element a ∈ G is the smallest positive integer n, such that an= e, where e denotes the identity element of the group, and an denotes the product of n copies of a.

## What is the order of S4?

(a) The possible cycle types of elements in S4 are: identity, 2-cycle, 3-cycle, 4- cycle, a product of two 2-cycles. These have orders 1, 2, 3, 4, 2 respectively, so the possible orders of elements in S4 are 1, 2, 3, 4.