What is the order of an alternating group of degree n?

What is the order of an alternating group of degree n?

The group An is abelian if and only if n ≤ 3 and simple if and only if n = 3 or n ≥ 5. A5 is the smallest non-abelian simple group, having order 60, and the smallest non-solvable group.

What is the order of the alternating group A4?

The group A4 has order 12, so its subgroups could have size 1, 2, 3, 4, 6, or 12. There are subgroups of orders 1, 2, 3, 4, and 12, but A4 has no subgroup of order 6 (equivalently, no subgroup of index 2). Here is one proof, using left cosets. Theorem 1.

What is the order of alternating group in?

Number of equivalence classes of various kinds

What is the order of alternating group A5?

Summary

Is the alternating group a normal subgroup?

alternating group is a normal subgroup of the symmetric group.

What is an alternating group of 4 elements?

A4 is the alternating group on 4 letters. That is it is the set of all even permutations. The elements are: (1),(12)(34),(13)(24),(14)(23),(123),(132),(124),(142),(134),(143),(234),(243)

What is the order of the alternating group A4 How many elements of order 2 are there in A4?

The group A4 has three elements of order 2, (1 2)(3 4), (1 3)(2 4), (1 4)(2 3).

Why is an alternating group Simple?

We call An the alternat ing group of degree n. A group is simple if it has no normal subgroups other that itself and 1. Indeed, any element of An is a product of transpositions of the form (ab)(cd) or (ab)(ac). Since (ab)(cd)=(acb) (acd) and (ab)(ac)=(acb) we conclude that An is generated by the 3-cycles.

What is the group A6?

The outer automorphism group of alternating group:A6 is a Klein four-group. In particular, it has order 4.

What is the symmetric group s4?

The symmetric group S4 is the group of all permutations of 4 elements. It has 4! =24 elements and is not abelian.