Elements of the Alternating Group

Prove that the number of elements in the alternating group $A_n$ is $\frac{n!}{2}$.

The problem of counting the elements in the alternating group is a fundamental aspect of group theory, particularly in understanding the structure and order of special types of subgroups within symmetric groups. The alternating group, denoted as $A_n$, consists of all the even permutations on a set of n elements. Understanding why the order of this group is n factorial divided by 2 requires knowledge of even and odd permutations and their relation to cycle decompositions within permutations.

A pivotal concept here is that permutations can be classified into even and odd classes based on their cycle structure. Specifically, even permutations can be constructed by an even number of transpositions, while odd permutations require an odd number of transpositions. The symmetric group $S_n$, which includes all permutations of n elements, naturally divides into two equal classes: even and odd. Hence, there are precisely n factorial permutations, half of which are even, leading to the order of the alternating group $A_n$.

In terms of strategy, proving this result often begins with establishing an understanding of even and odd permutations. One could explore transpositions, understand their properties alongside properties of the symmetric group, and utilize Lagrange's theorem to effectively reveal the index and order of $A_n$. This problem typically serves as a springboard to broader concepts in group theory, such as the study of normal subgroups and the characteristics of simpler group structures within the larger symmetric group context.