For a group to be of order N means that it has N elements. This is also referred to as the group's cardinality or size.
Showing that S_N is of order N is important because it helps us understand the structure and properties of symmetric groups. It also allows us to make connections between different mathematical concepts and applications.
To prove that S_N is of order N, one can use the definition of symmetric groups and show that it contains N elements. This can be done through various methods, such as constructing the group's multiplication table or using mathematical induction.
Understanding the order of symmetric groups has many practical applications in fields such as cryptography, combinatorics, and physics. For example, in cryptography, symmetric groups are used in the development of secure encryption algorithms.
No, S_N can only be of order N. This is because the number of elements in a symmetric group is directly related to the number of elements in the set that it is permuting, which is always N in the case of S_N.