- #1
Azure Ace
How can you prove that there can only be 2 possible four-element group?
fresh_42 said:This is basically correct. An element of order ##4## leads to ##\mathbb{Z}_4##. So there is only the possibility left, where we have only elements of order ##2##. Basically this can lead to the group ##\mathbb{Z}_2^2## and groups ##\mathbb{Z}_2 \rtimes_\varphi \mathbb{Z}_2##. Now we need an argument, why ##\varphi = 1## is the only possibility, i.e. why all semidirect products are already direct.
To your second question: What do you mean by "one element order"? If you mean, whether all group elements have to be of the same order, then the answer is no. E.g. ##\mathbf{Sym}(3) = \langle (1),(12),(123) \rangle## has elements of order ##2## and ##3##. Also ##1 \in \langle \mathbb{Z}_4 , + \rangle## is of order ##4## whereas ##2 \in \langle \mathbb{Z}_4 , + \rangle## is of order ##2##
The number of possible four-element groups is determined by the order of the elements, not the elements themselves. In other words, the same four elements can be arranged in different orders, resulting in different groups. However, there are only two possible ways to arrange four elements - in a group of two pairs or in a group of four individual elements. This is due to the mathematical concept of permutation.
No, there cannot be more than two four-element groups. As mentioned before, the number of possible groups is determined by the order of the elements, and there are only two ways to arrange four elements.
No, the two four-element groups are not the same. While they both have four elements, they are arranged in different ways and therefore represent different mathematical concepts and operations.
The concept of permutation applies to any number of elements, not just four. For example, with three elements, there are six possible ways to arrange them (ABC, ACB, BAC, BCA, CAB, CBA). With five elements, there are 120 possible arrangements.
Understanding the number of possible four-element groups is important in mathematics as it relates to the concept of permutation and helps to explain why certain mathematical operations behave the way they do. It also has practical applications in fields such as computer science and cryptography.