Abstract Algebra:Permutations

Question

1)Express the permutations $\alpha=(24)\in S_4$ and $\beta=(1)\in S_5$, as sets. a) Describe the permutations which are reflexive b) What types of permutations are partial orders.

Attempt of a solution I understand that the first permutation is $\{(1,1),(2,4),(3,3),(4,4)\}$ and the second permutation is $\{(1,1),(2,2),(3,3),(4,4),(5,5)\}$ . a)Is it sufficient to say A reflexive permutations is (1), (2),(3),(4) for the first part and describe the second part similarly? b) is it sufficient to say the permutation must be reflexive, antisymmetric and transitive

Answer 1

It seems these permutations are intended to be expressed as relations. E.g. $(123) \in S_4$ would be equivalent to the relation $\{(1,2),(2,3),(3,1),(4,4)\}$. In this example, this is a way of writing:

• $1$ maps to $2$,
• $2$ maps to $3$,
• $3$ maps to $1$, and
• $4$ maps to $4$ (a fixed point).

Note that the domain and range are both $\{1,2,3,4\}$ since this is a permutation.

Your task is to do this for $(24) \in S_4$ which has the fixed points $1$ and $3$, and for $(1) \in S_5$, which I think is the identity permutation (i.e., all points are fixed). In the second case, the domain and range will instead be $\{1,2,3,4,5\}$.

For the second part:

• A reflexive relation contains $(i,i)$ for all $i$ in its domain. Basically this tells you where everything is mapped to.
• A partial order is reflexive (and antisymmetric and transitive).