Publish On: 2019-05-17

# Ben joy

Total Post: 542

## Question: Define the orbit of a permutation

Reply On: 2013-10-07

# Mjay Jollay

Total Post: 0

## ANS: Define the orbit of a permutation

Let ƒ be any permutation on a set S. If a relation ~ is defined on S such that a ~ b ⟺ ƒ

(i) Reflexive, because a ~ a ƒ

(ii) Symmetric, because a ~ b ƒ

a = ƒ

(iii) Transitive, because a ~ b and b ~ c

ƒ

ƒ

ƒ

Thus above defined relation ~ is an equivalence relation on S and hence it partitions S into mutually disjoint classes. Each equivalence class determined by the above relation is called an orbit of ƒ.

^{(n)}(a) = b for some integer n and ∀ a, b S, we observe that the relation ‘~’ is(i) Reflexive, because a ~ a ƒ

^{(n)}(a) = I (a) = a ∀ a S.(ii) Symmetric, because a ~ b ƒ

^{(n)}(a) = b for some integer na = ƒ

^{(n)}(b) b – a for a, b S.(iii) Transitive, because a ~ b and b ~ c

ƒ

^{(n)}(a) = b, ƒ^{(m)}(b) = c for some integers n and m.ƒ

^{m}(ƒ^{n}(a)) = ƒ^{m}(b) = cƒ

^{m+n}(a) = c for some integer (m + n) a ~ c.Thus above defined relation ~ is an equivalence relation on S and hence it partitions S into mutually disjoint classes. Each equivalence class determined by the above relation is called an orbit of ƒ.

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