# Finite simple group

This page describes a group property obtained as a conjunction (AND) of two (or more) more fundamental group properties: finite group and simple group
View other group property conjunctions OR view all group properties

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## Definition

### Symbol-free definition

A group is said to be a finite simple group if it satisfies both these conditions:

## Related notions

### Classification problem

The Classification problem is the problem of obtaining a complete description or classification of all the finite simple groups. The Classification Problem has now been solved, viz it has been shown that every finite simple group is one of these:

Further information: Classification of finite simple groups

### Feasibility

A natural number is said to be simple-feasible if it occurs as the order of a simple group. One of the important initial questions in the classification of simple groups is: what are the simple-feasible numbers?

To answer this question, we need a mix of techniques from Sylow theory, representation theory, group actions, and of course basic arguments with subgroups.

Further information: simple-feasible number