# Finite group

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This article defines a group property that is pivotal (i.e., important) among existing group properties
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## Definition

A group $G$ is said to be finite if the cardinality of its underlying set (i.e., its order) is finite. Here, the cardinality of a set refers to the number of elements in the set, and is denoted as $|G|$.

## Examples

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The trivial group is an example of a finite group -- the underlying set has cardinality one. Other examples of finite groups include the symmetric group on a set, and the cyclic group of order $n$. Any subgroup of a finite group is finite.

The group of integers, group of rational numbers, and group of real numbers (each under addition) are not finite groups.

## 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
• order of element divides order of group
• order of quotient group divides order of 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.

Conversely, if the lattice of subgroups of a group is finite, then the group itself is finite. For a proof, see finitely many subgroups iff finite

## Metaproperties

Metaproperty name Satisfied? Proof Statement with symbols
subgroup-closed group property Yes Suppose $G$ is a finite group and $H$ is a subgroup of $G$. Then, $H$ is also a finite group.
quotient-closed group property Yes Suppose $G$ is a finite group and $H$ is a normal subgroup of $G$. Then, the quotient group $G/H$ is also a finite group.
finite direct product-closed group property Yes Suppose $G_1, G_2, \dots, G_n$ are finite groups. Then, the direct product $G_1 \times G_2 \times \dots \times G_n$ is also a finite group.
lattice-determined group property Yes finitely many subgroups iff finite Whether or not a group is finite is determined completely from its lattice of subgroups. Specifically, a group $G$ is finite if and only if its lattice of subgroups, $L(G)$ is finite. Therefore if two groups $G_1, G_2$ have isomorphic lattices of subgroups, either both are finite, or neither is.

## 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

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### Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
group of prime power order finite group whose order is a power of a prime number (by definition) |FULL LIST, MORE INFO
odd-order group finite group whose order is an odd number (by definition) |FULL LIST, MORE INFO

### Conjunction with other properties

Conjunction Other component of conjunction Intermediate notions between finite group and conjunction Intermediate notions between other component and conjunction