Finite group

Symbol-free definition
A group is said to be finite if the cardinality of its underlying set (viz its order) is finite. Here, the cardinality of a set refers to the number of elements in the set.

Definition with symbols
A group $$G$$ is finite if the cardinality of the set $$G$$ is finite. In other words, $$G$$ has only finitely many elements.

Examples
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.

Relation with other properties
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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.