Finite group

This article is about a basic definition in group theory. The article text may, however, contain advanced material.
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This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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Definition

Symbol-free definition

A group is said to be finite if the cardinality of its underlying set (viz its order) is finite.

Definition with symbols

A group $G$ is finite if the cardinality of the set $G$ is finite.

Relation with other properties

This property is a pivotal (important) member of its property space. Its variations, opposites, and other properties related to it and defined using it are often studied

Some of these can be found at:

• Category: Variations of finiteness (groups)
• Category: Group properties tautological for finite groups
• Category: Group properties giving finiteness condition

Facts

Monoid generated is same as subgroup generated

In a finite group, the monoid generated by any subset is the same as the subgroup generated by it. This follows from the fact that since every element in a finite group has finite order, the inverse of any element can be written as a power of that element.

Theorems on order-dividing

When we are working in finite groups, we can use results like these:

• Lagrange's theorem states that the order of any subgroup divides the order of the group
• Also, the order of any quotient group divides the order of the group
• Sylow's theorem tells us that for any prime $p$, there exist $p$-Sylow subgroups, viz $p$-subgroups whose index is relatively prime to $p$.

Existence of minimal and maximal elements

The lattice of subgroups of a finite group is a finite lattice, hence we can locate minimal elements and maximal elements, and do other things like find a finite stage at which every ascending/descending chain stabilizes.