Clarification for Alternating Group

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In summary, we were discussing in class how to show that ${A}_{5}$ is cyclic. We determined that in order for ${A}_{5}$ to be cyclic, there must exist an element $\alpha$ in ${A}_{5}$ such that $<\alpha> = {A}_{5}$ and $ord(\alpha) = |<\alpha>| = |{A}_{5}| = \frac{5!}{2} = 60$. However, we then realized that the product of a 4-cycle, 3-cycle, and 5-cycle would result in an odd number of 2-cycles, making it not a part of ${A}_{5}$. Therefore, we concluded that ${A}_{5
  • #1
Enzipino
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In class we had to show that ${A}_{5}$ is cyclic. So what we did was,

${A}_{5}$ is cyclic iff there is an $\alpha\in{A}_{5}$ with $<\alpha> = {A}_{5}$. So, the $ord(\alpha) = |<\alpha>| = |{A}_{5}| = \frac{5!}{2} = 60$. So, $60 = {2}^{2}*3*5$.

After this, we said that we could do a 4-cycle, 3-cycle, and 5-cycle, which would be in ${S}_{12}$ but not ${A}_{5}$. We concluded by saying ${A}_{5}$ is not cyclic.

The main thing I'm confused about is why the 4-cycle, 3-cycle, and 5-cycle is not in ${A}_{5}$. Could someone just clarify this?
 
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  • #2
Hi Enzipino,

I'm not sure what you are asking for, but I think it is why the product of a 3-cycle, a 4-cycle and a 5-cycle can't be in the alternating group.

A $n-$cycle can be decomposed as the product of $n-1$ 2-cycles, hence you got the product of $2+3+4=9$ 2-cycles, which is odd, so it is not in the alternating group.
 

FAQ: Clarification for Alternating Group

What is the alternating group?

The alternating group, denoted as An, is a type of mathematical group that consists of even permutations of a set of n elements. It is a subgroup of the symmetric group, Sn, which contains all possible permutations of the n elements.

How is the alternating group different from the symmetric group?

The alternating group only contains even permutations, while the symmetric group contains both even and odd permutations. Additionally, the alternating group is a normal subgroup of the symmetric group, meaning that it is closed under the group operation of composition.

What are the properties of the alternating group?

Some key properties of the alternating group include: it is a finite group, it is non-abelian (meaning that the order of composition matters), it is simple (meaning it has no non-trivial normal subgroups), and it is a solvable group (meaning that it can be solved by a sequence of subgroups).

How is the alternating group used in mathematics?

The alternating group has many applications in mathematics, particularly in group theory and abstract algebra. It is used to study symmetry and patterns, as well as in combinatorics and number theory. It also has applications in physics, specifically in quantum mechanics and molecular symmetry.

Can you provide an example of the alternating group?

Yes, an example of the alternating group is A4, which consists of even permutations of 4 elements. It has 12 elements: (1), (12)(34), (13)(24), (14)(23), (123), (132), (124), (142), (134), (143), (234), (243).

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