## How do you classify finite groups?

The classification of finite simple groups is a theorem stating that every finite simple group belongs to one of the following families:

- A cyclic group with prime order;
- An alternating group of degree at least 5;
- A simple group of Lie type;
- One of the 26 sporadic simple groups;

### How long is the classification of finite simple groups?

It is expected that they will number about twelve, and stretch to a total of 3,000-4,000 pages. At time of writing, 6 have been published so far. A second major theme in finite group theory is the extension problem for finite groups.

**What is the most important theorem on finite groups?**

The most important structure theorem for finite groups is the Jordan–Hölder Theorem, which shows that any finite group is built up from finite simple groups.

**Are all finite groups classified?**

Simple groups can be seen as the basic building blocks of all finite groups, reminiscent of the way the prime numbers are the basic building blocks of the natural numbers….Timeline of the proof.

Publication date | |
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1974 | Thompson classifies N-groups, groups all of whose local subgroups are solvable. |

## What are the classification of groups?

Classification of Groups

- Primary and Secondary Groups.
- Membership and Reference Groups.
- Small and Large Groups.
- Organized and Unorganized Groups.
- In and Out-going Groups.
- Accidental and Purposive Groups.
- Open and Closed Groups.
- Temporary and Permanent Groups.

### What is finite and infinite group?

An infinite set is endless from the start or end, but both the side could have continuity unlike in Finite set where both start and end elements are there. If a set has the unlimited number of elements, then it is infinite and if the elements are countable then it is finite.

**What is a group classification?**

Group can be classified into many forms. Basically, groups are classified as formal and informal. A formal group can be divided into the Command group and Task group. Command Group: A command group is determining the organizational chart. It is composed of the individuals who report directly to a given manager.

**Are all finite groups cyclic?**

Every cyclic group is virtually cyclic, as is every finite group. An infinite group is virtually cyclic if and only if it is finitely generated and has exactly two ends; an example of such a group is the direct product of Z/nZ and Z, in which the factor Z has finite index n.

## What is the order of 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.

### Can a finite group have elements of infinite order?

No, because in a finite group every element has its order less than equal to the order of group. let G be a finite group of order n. It cannot have an element of infinite order since a belongs to G implies a^n=e and so o(a) is any positive integer less than or equal to n.

**Are finite groups cyclic?**

**What are the 4 types of groups?**

Four basic types of groups have traditionally been recognized: primary groups, secondary groups, collective groups, and categories.

## What is a finite group?

Finite group. In abstract algebra, a finite group is a mathematical group with a finite number of elements. A group is a set of elements together with an operation which associates, to each ordered pair of elements, an element of the set. In the case of a finite group, the set is finite.

### What is representation theory?

Representation theory. Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures.

**What is Group classification?**

Group classification is determined by member’s occupation, position, or job duties. State law classifies types of employment within groups and NCRS assign members to the group classification for their position.