Alternating Group

Dependencies:

1. Parity of a permutation
2. Condition for a subset to be a subgroup
3. Inverse of product of two elements of a group

The alternating group $A_X$ is the set of all even permutations of $X$.

Proof that $A_X$ is a group

Let $\sigma_1 = \pi_1\pi_2\ldots\pi_{2m}$ and $\sigma_2 = \tau_1\tau_2\ldots\tau_{2n}$ be even permutations. \[ \sigma_1\sigma_2^{-1} = \pi_1\pi_2\ldots\pi_{2m}(\tau_1\tau_2\ldots\tau_{2n})^{-1} = \pi_1\pi_2\ldots\pi_{2m}\tau_{2n}^{-1}\tau_{2n-1}^{-1}\ldots\tau_1^{-1} = \pi_1\pi_2\ldots\pi_{2m}\tau_{2n}\tau_{2n-1}\ldots\tau_1 \in A_X \]

Therefore, $A_X$ is a subgroup of $S_X$.

Dependency for: None

Info:

• Depth: 6
• Number of transitive dependencies: 15

Transitive dependencies:

1. /sets-and-relations/relation-composition-is-associative
2. /sets-and-relations/composition-of-bijections-is-a-bijection
3. Group
4. Inverse of product of two elements of a group
5. Identity of a group is unique
6. Subgroup
7. Permutation group
8. Product of cycles and a transposition
9. Permutation is disjoint cycle product
10. Product of disjoint cycles is commutative
11. Canonical cycle notation of a permutation
12. Parity of a permutation
13. Inverse of a group element is unique
14. Conditions for a subset to be a subgroup
15. Condition for a subset to be a subgroup