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$$\text{ Let } n∈ \mathbb{N} \text{ and } S_{n} \text{ symmetrical group on } \underline n\underline .

\text{ Let }

π ∈ S_{n} \text{ and z } \text{ the number of disjunctive Cycles of π. Here will be counted 1 - Cycle }. (a) \text{ Prove that } sgn (π) = (-1)^{n-z}.

(b) \text{ Prove that subset } A_{n}= \{π∈Sn∣sgn(π)=1\} ⊆ S_{n}\text{ is subgroup of } S_{n}.

(c)

\text{ Find number of elements } |A_{n}| \text{ of a subgroup } A_{n} \text{ from (b) }

$$

\text{ Let }

π ∈ S_{n} \text{ and z } \text{ the number of disjunctive Cycles of π. Here will be counted 1 - Cycle }. (a) \text{ Prove that } sgn (π) = (-1)^{n-z}.

(b) \text{ Prove that subset } A_{n}= \{π∈Sn∣sgn(π)=1\} ⊆ S_{n}\text{ is subgroup of } S_{n}.

(c)

\text{ Find number of elements } |A_{n}| \text{ of a subgroup } A_{n} \text{ from (b) }

$$

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